# lfunc_search downloaded from the LMFDB on 05 April 2026. # Search link: https://www.lmfdb.org/L/1/236/236.155 # Query "{'degree': 1, 'conductor': 236}" returned 57 lfunc_searchs, sorted by root analytic conductor. # Each entry in the following data list has the form: # [Label, $\alpha$, $A$, $d$, $N$, $\chi$, $\mu$, $\nu$, $w$, prim, arith, $\mathbb{Q}$, self-dual, $\operatorname{Arg}(\epsilon)$, $r$, First zero, Origin] # For more details, see the definitions at the bottom of the file. "1-236-236.103-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.103" [[0, 0.0]] [] 0 true true false false -0.286157081499911 0 1.11462792572 ["Character/Dirichlet/236/103"] "1-236-236.11-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.11" [[0, 0.0]] [] 0 true true false false 0.1879453904544102 0 1.87063299108 ["Character/Dirichlet/236/11"] "1-236-236.111-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.111" [[0, 0.0]] [] 0 true true false false -0.03138347243325436 0 1.62479779432 ["Character/Dirichlet/236/111"] "1-236-236.115-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.115" [[0, 0.0]] [] 0 true true false false 0.14268458902573322 0 1.61033083517 ["Character/Dirichlet/236/115"] "1-236-236.131-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.131" [[0, 0.0]] [] 0 true true false false -0.021265642470344456 0 0.855375134463 ["Character/Dirichlet/236/131"] "1-236-236.151-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.151" [[0, 0.0]] [] 0 true true false false -0.0596110070460603 0 1.1282250352 ["Character/Dirichlet/236/151"] "1-236-236.155-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.155" [[0, 0.0]] [] 0 true true false false -0.30141932511299446 0 0.811431403363 ["Character/Dirichlet/236/155"] "1-236-236.179-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.179" [[0, 0.0]] [] 0 true true false false 0.025557814089232155 0 1.12037285207 ["Character/Dirichlet/236/179"] "1-236-236.183-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.183" [[0, 0.0]] [] 0 true true false false -0.44253776107782217 0 2.29290586384 ["Character/Dirichlet/236/183"] "1-236-236.187-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.187" [[0, 0.0]] [] 0 true true false false 0.44253776107782217 0 0.151386847004 ["Character/Dirichlet/236/187"] "1-236-236.191-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.191" [[0, 0.0]] [] 0 true true false false -0.1674852836364173 0 1.20076552783 ["Character/Dirichlet/236/191"] "1-236-236.195-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.195" [[0, 0.0]] [] 0 true true false false -0.28165038314115876 0 0.528622157748 ["Character/Dirichlet/236/195"] "1-236-236.207-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.207" [[0, 0.0]] [] 0 true true false false -0.025557814089232155 0 1.46895540421 ["Character/Dirichlet/236/207"] "1-236-236.211-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.211" [[0, 0.0]] [] 0 true true false false 0.0596110070460603 0 1.73748058041 ["Character/Dirichlet/236/211"] "1-236-236.215-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.215" [[0, 0.0]] [] 0 true true false false 0.1674852836364173 0 1.89042663007 ["Character/Dirichlet/236/215"] "1-236-236.219-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.219" [[0, 0.0]] [] 0 true true false false 0.03138347243325436 0 1.82789021519 ["Character/Dirichlet/236/219"] "1-236-236.227-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.227" [[0, 0.0]] [] 0 true true false false 0.021265642470344456 0 1.21718707699 ["Character/Dirichlet/236/227"] "1-236-236.23-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.23" [[0, 0.0]] [] 0 true true false false 0.28165038314115876 0 1.46223941505 ["Character/Dirichlet/236/23"] "1-236-236.231-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.231" [[0, 0.0]] [] 0 true true false false 0.2363730068189395 0 1.50618497712 ["Character/Dirichlet/236/231"] "1-236-236.235-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.235" [[0, 0.0]] [] 0 true true true true 0.0 0 1.10287633413 ["Character/Dirichlet/236/235"] "1-236-236.31-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.31" [[0, 0.0]] [] 0 true true false false -0.2716075042445951 0 0.898787855253 ["Character/Dirichlet/236/31"] "1-236-236.39-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.39" [[0, 0.0]] [] 0 true true false false -0.14268458902573322 0 0.801434770573 ["Character/Dirichlet/236/39"] "1-236-236.43-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.43" [[0, 0.0]] [] 0 true true false false -0.1879453904544102 0 0.927499497032 ["Character/Dirichlet/236/43"] "1-236-236.47-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.47" [[0, 0.0]] [] 0 true true false false -0.2363730068189395 0 0.257718031499 ["Character/Dirichlet/236/47"] "1-236-236.55-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.55" [[0, 0.0]] [] 0 true true false false 0.286157081499911 0 2.04023691946 ["Character/Dirichlet/236/55"] "1-236-236.67-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.67" [[0, 0.0]] [] 0 true true false false 0.30141932511299446 0 2.19664392959 ["Character/Dirichlet/236/67"] "1-236-236.83-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.83" [[0, 0.0]] [] 0 true true false false 0.31371293560140356 0 1.92806263908 ["Character/Dirichlet/236/83"] "1-236-236.91-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.91" [[0, 0.0]] [] 0 true true false false -0.31371293560140356 0 0.745546501369 ["Character/Dirichlet/236/91"] "1-236-236.99-r0-0-0" 1.0959793924061765 1.0959793924061765 1 236 "236.99" [[0, 0.0]] [] 0 true true false false 0.2716075042445951 0 1.90523436598 ["Character/Dirichlet/236/99"] "1-236-236.107-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.107" [[1, 0.0]] [] 0 true true false false 0.10263490709291821 0 1.1681919901 ["Character/Dirichlet/236/107"] "1-236-236.123-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.123" [[1, 0.0]] [] 0 true true false false -0.08414568668903227 0 0.702183958294 ["Character/Dirichlet/236/123"] "1-236-236.127-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.127" [[1, 0.0]] [] 0 true true false false 0.019471783124785747 0 0.722028434373 ["Character/Dirichlet/236/127"] "1-236-236.135-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.135" [[1, 0.0]] [] 0 true true false false -0.043466586833824754 0 1.19411765631 ["Character/Dirichlet/236/135"] "1-236-236.139-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.139" [[1, 0.0]] [] 0 true true false false 0.026172875153663445 0 1.04794791435 ["Character/Dirichlet/236/139"] "1-236-236.143-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.143" [[1, 0.0]] [] 0 true true false false 0.35912387185404515 0 1.9000761007 ["Character/Dirichlet/236/143"] "1-236-236.147-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.147" [[1, 0.0]] [] 0 true true false false -0.4822019214558697 0 0.249567602161 ["Character/Dirichlet/236/147"] "1-236-236.15-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.15" [[1, 0.0]] [] 0 true true false false 0.45775750583420854 0 0.433911670884 ["Character/Dirichlet/236/15"] "1-236-236.159-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.159" [[1, 0.0]] [] 0 true true false false 0.3166898409956339 0 1.0394799783 ["Character/Dirichlet/236/159"] "1-236-236.163-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.163" [[1, 0.0]] [] 0 true true false false -0.026172875153663445 0 1.27278713579 ["Character/Dirichlet/236/163"] "1-236-236.167-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.167" [[1, 0.0]] [] 0 true true false false -0.3672272304270158 0 0.571848612368 ["Character/Dirichlet/236/167"] "1-236-236.171-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.171" [[1, 0.0]] [] 0 true true false false 0.3672272304270158 0 1.3538397892 ["Character/Dirichlet/236/171"] "1-236-236.175-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.175" [[1, 0.0]] [] 0 true true false false 0.4822019214558697 0 1.63989843339 ["Character/Dirichlet/236/175"] "1-236-236.19-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.19" [[1, 0.0]] [] 0 true true false false 0.4615142550030228 0 1.79683806155 ["Character/Dirichlet/236/19"] "1-236-236.199-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.199" [[1, 0.0]] [] 0 true true false false 0.48901328766409574 0 0.332093466975 ["Character/Dirichlet/236/199"] "1-236-236.203-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.203" [[1, 0.0]] [] 0 true true false false -0.35912387185404515 0 0.253553999848 ["Character/Dirichlet/236/203"] "1-236-236.223-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.223" [[1, 0.0]] [] 0 true true false false -0.019471783124785747 0 0.451918897625 ["Character/Dirichlet/236/223"] "1-236-236.27-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.27" [[1, 0.0]] [] 0 true true false false 0.1674720176516187 0 1.34345732873 ["Character/Dirichlet/236/27"] "1-236-236.3-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.3" [[1, 0.0]] [] 0 true true false false 0.15264926306456977 0 1.29706101324 ["Character/Dirichlet/236/3"] "1-236-236.35-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.35" [[1, 0.0]] [] 0 true true false false -0.1674720176516187 0 0.398163406049 ["Character/Dirichlet/236/35"] "1-236-236.51-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.51" [[1, 0.0]] [] 0 true true false false -0.48901328766409574 0 2.17917191034 ["Character/Dirichlet/236/51"] "1-236-236.63-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.63" [[1, 0.0]] [] 0 true true false false -0.45775750583420854 0 2.24171252226 ["Character/Dirichlet/236/63"] "1-236-236.7-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.7" [[1, 0.0]] [] 0 true true false false 0.043466586833824754 0 1.52441917171 ["Character/Dirichlet/236/7"] "1-236-236.71-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.71" [[1, 0.0]] [] 0 true true false false 0.08414568668903227 0 0.614023733538 ["Character/Dirichlet/236/71"] "1-236-236.75-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.75" [[1, 0.0]] [] 0 true true false false -0.10263490709291821 0 0.874913985393 ["Character/Dirichlet/236/75"] "1-236-236.79-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.79" [[1, 0.0]] [] 0 true true false false -0.15264926306456977 0 0.666447220027 ["Character/Dirichlet/236/79"] "1-236-236.87-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.87" [[1, 0.0]] [] 0 true true false false -0.4615142550030228 0 0.355459638957 ["Character/Dirichlet/236/87"] "1-236-236.95-r1-0-0" 25.361722251531504 25.361722251531504 1 236 "236.95" [[1, 0.0]] [] 0 true true false false -0.3166898409956339 0 0.052946750545 ["Character/Dirichlet/236/95"] # Label -- # Each L-function $L$ has a label of the form d-N-q.k-x-y-i, where # * $d$ is the degree of $L$. # * $N$ is the conductor of $L$. When $N$ is a perfect power $m^n$ we write $N$ as $m$e$n$, since $N$ can be very large for some imprimitive L-functions. # * q.k is the label of the primitive Dirichlet character from which the central character is induced. # * x-y is the spectral label encoding the $\mu_j$ and $\nu_j$ in the analytically normalized functional equation. # * i is a non-negative integer disambiguating between L-functions that would otherwise have the same label. #$\alpha$ (root_analytic_conductor) -- # If $d$ is the degree of the L-function $L(s)$, the **root analytic conductor** $\alpha$ of $L$ is the $d$th root of the analytic conductor of $L$. It plays a role analogous to the root discriminant for number fields. #$A$ (analytic_conductor) -- # The **analytic conductor** of an L-function $L(s)$ with infinity factor $L_{\infty}(s)$ and conductor $N$ is the real number # \[ # A := \mathrm{exp}\left(2\mathrm{Re}\left(\frac{L_{\infty}'(1/2)}{L_{\infty}(1/2)}\right)\right)N. # \] #$d$ (degree) -- # The **degree** of an L-function is the number $J + 2K$ of Gamma factors occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The degree appears as the first component of the Selberg data of $L(s).$ In all known cases it is the degree of the polynomial of the inverse of the Euler factor at any prime not dividing the conductor. #$N$ (conductor) -- # The **conductor** of an L-function is the integer $N$ occurring in its functional equation # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s). # \] # The conductor of an analytic L-function is the second component in the Selberg data. For a Dirichlet L-function # associated with a primitive Dirichlet character, the conductor of the L-function is the same as the conductor of the character. For a primitive L-function associated with a cusp form $\phi$ on $GL(2)/\mathbb Q$, the conductor of the L-function is the same as the level of $\phi$. # In the literature, the word _level_ is sometimes used instead of _conductor_. #$\chi$ (central_character) -- # An L-function has an Euler product of the form # $L(s) = \prod_p L_p(p^{-s})^{-1}$ # where $L_p(x) = 1 + a_p x + \ldots + (-1)^d \chi(p) x^d$. The character $\chi$ is a Dirichlet character mod $N$ and is called **central character** of the L-function. # Here, $N$ is the conductor of $L$. #$\mu$ (mus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$\nu$ (nus) -- # All known analytic L-functions have a **functional equation** that can be written in the form # \[ # \Lambda(s) := N^{s/2} # \prod_{j=1}^J \Gamma_{\mathbb R}(s+\mu_j) \prod_{k=1}^K \Gamma_{\mathbb C}(s+\nu_k) # \cdot L(s) = \varepsilon \overline{\Lambda}(1-s), # \] # where $N$ is an integer, $\Gamma_{\mathbb R}$ and $\Gamma_{\mathbb C}$ are defined in terms of the $\Gamma$-function, $\mathrm{Re}(\mu_j) = 0 \ \mathrm{or} \ 1$ (assuming Selberg's eigenvalue conjecture), and $\mathrm{Re}(\nu_k)$ is a positive integer # or half-integer, # \[ # \sum \mu_j + 2 \sum \nu_k \ \ \ \ \text{is real}, # \] # and $\varepsilon$ is the sign of the functional equation. # With those restrictions on the spectral parameters, the # data in the functional equation is specified uniquely. The integer $d = J + 2 K$ # is the degree of the L-function. The integer $N$ is the conductor (or level) # of the L-function. The pair $[J,K]$ is the signature of the L-function. The parameters # in the functional equation can be used to make up the 4-tuple called the Selberg data. # The axioms of the Selberg class are less restrictive than # given above. # Note that the functional equation above has the central point at $s=1/2$, and relates $s\leftrightarrow 1-s$. # For many L-functions there is another normalization which is natural. The corresponding functional equation relates $s\leftrightarrow w+1-s$ for some positive integer $w$, # called the motivic weight of the L-function. The central point is at $s=(w+1)/2$, and the arithmetically normalized Dirichlet coefficients $a_n n^{w/2}$ are algebraic integers. #$w$ (motivic_weight) -- # The **motivic weight** (or **arithmetic weight**) of an arithmetic L-function with analytic normalization $L_{an}(s)=\sum_{n=1}^\infty a_nn^{-s}$ is the least nonnegative integer $w$ for which $a_nn^{w/2}$ is an algebraic integer for all $n\ge 1$. # If the L-function arises from a motive, then the weight of the motive has the # same parity as the motivic weight of the L-function, but the weight of the motive # could be larger. This apparent discrepancy comes from the fact that a Tate twist # increases the weight of the motive. This corresponds to the change of variables # $s \mapsto s + j$ in the L-function of the motive. #prim (primitive) -- # An L-function is primitive if it cannot be written as a product of nontrivial L-functions. The "trivial L-function" is the constant function $1$. #arith (algebraic) -- # An L-function $L(s) = \sum_{n=1}^{\infty} a_n n^{-s}$ is called **arithmetic** if its Dirichlet coefficients $a_n$ are algebraic numbers. #$\mathbb{Q}$ (rational) -- # A **rational** L-function $L(s)$ is an arithmetic L-function with coefficient field $\Q$; equivalently, its Euler product in the arithmetic normalization can be written as a product over rational primes # \[ # L(s)=\prod_pL_p(p^{-s})^{-1} # \] # with $L_p\in \Z[T]$. #self-dual (self_dual) -- # An L-function $L(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$ is called **self-dual** if its Dirichlet coefficients $a_n$ are real. #$\operatorname{Arg}(\epsilon)$ (root_angle) -- # The **root angle** of an L-function is the argument of its root number, as a real number $\alpha$ with $-0.5 < \alpha \le 0.5$. #$r$ (order_of_vanishing) -- # The **analytic rank** of an L-function $L(s)$ is its order of vanishing at its central point. # When the analytic rank $r$ is positive, the value listed in the LMFDB is typically an upper bound that is believed to be tight (in the sense that there are known to be $r$ zeroes located very near to the central point). #First zero (z1) -- # The **zeros** of an L-function $L(s)$ are the complex numbers $\rho$ for which $L(\rho)=0$. # Under the Riemann Hypothesis, every non-trivial zero $\rho$ lies on the critical line $\Re(s)=1/2$ (in the analytic normalization). # The **lowest zero** of an L-function $L(s)$ is the least $\gamma>0$ for which $L(1/2+i\gamma)=0$. Note that even when $L(1/2)=0$, the lowest zero is by definition a positive real number. #Origin (instance_urls) -- # L-functions arise from many different sources. Already in degree 2 we have examples of # L-functions associated with holomorphic cusp forms, with Maass forms, with elliptic curves, with characters of number fields (Hecke characters), and with 2-dimensional representations of the Galois group of a number field (Artin L-functions). # Sometimes an L-function may arise from more than one source. For example, the L-functions associated with elliptic curves are also associated with weight 2 cusp forms. A goal of the Langlands program ostensibly is to prove that any degree $d$ L-function is associated with an automorphic form on $\mathrm{GL}(d)$. Because of this representation theoretic genesis, one can associate an L-function not only to an automorphic representation but also to symmetric powers, or exterior powers of that representation, or to the tensor product of two representations (the Rankin-Selberg product of two L-functions).