# lfunc_search downloaded from the LMFDB on 25 May 2026. # Search link: https://www.lmfdb.org/L/1/161/161.103/r0-0 # Query "{'degree': 1, 'conductor': 161, 'spectral_label': 'r0-0'}" returned 53 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. "1-161-161.10-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.10" [[0, 0.0]] [] 0 true true false false 0.0014245432291719385 0 1.23642693605 ["Character/Dirichlet/161/10"] "1-161-161.100-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.100" [[0, 0.0]] [] 0 true true false false -0.03709068545967084 0 1.05741005514 ["Character/Dirichlet/161/100"] "1-161-161.103-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.103" [[0, 0.0]] [] 0 true true false false 0.2629859074412635 0 1.30592128929 ["Character/Dirichlet/161/103"] "1-161-161.111-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.111" [[0, 0.0]] [] 0 true true false false 0.3884776822344603 0 1.64871255879 ["Character/Dirichlet/161/111"] "1-161-161.121-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.121" [[0, 0.0]] [] 0 true true false false 0.05195358841774542 0 0.605385314034 ["Character/Dirichlet/161/121"] "1-161-161.122-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.122" [[0, 0.0]] [] 0 true true false false 0.038921362878720306 0 1.99210984554 ["Character/Dirichlet/161/122"] "1-161-161.123-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.123" [[0, 0.0]] [] 0 true true false false -0.22393030518028365 0 1.44986903177 ["Character/Dirichlet/161/123"] "1-161-161.125-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.125" [[0, 0.0]] [] 0 true true false false 0.4812515901752258 0 1.7879875866 ["Character/Dirichlet/161/125"] "1-161-161.128-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.128" [[0, 0.0]] [] 0 true true false false -0.06379682108944211 0 1.44438566232 ["Character/Dirichlet/161/128"] "1-161-161.129-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.129" [[0, 0.0]] [] 0 true true false false 0.13103787811379736 0 1.00781207637 ["Character/Dirichlet/161/129"] "1-161-161.132-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.132" [[0, 0.0]] [] 0 true true false false -0.3884776822344603 0 0.802866696273 ["Character/Dirichlet/161/132"] "1-161-161.136-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.136" [[0, 0.0]] [] 0 true true false false -0.2629859074412635 0 0.574130428886 ["Character/Dirichlet/161/136"] "1-161-161.142-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.142" [[0, 0.0]] [] 0 true true false false 0.1348860313028674 0 2.09862212048 ["Character/Dirichlet/161/142"] "1-161-161.143-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.143" [[0, 0.0]] [] 0 true true false false -0.09069197200590509 0 1.36623265051 ["Character/Dirichlet/161/143"] "1-161-161.144-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.144" [[0, 0.0]] [] 0 true true false false -0.1348860313028674 0 1.82289299808 ["Character/Dirichlet/161/144"] "1-161-161.145-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.145" [[0, 0.0]] [] 0 true true false false 0.0014245432291719385 0 1.1868141007 ["Character/Dirichlet/161/145"] "1-161-161.151-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.151" [[0, 0.0]] [] 0 true true false false -0.24936355918994513 0 1.26748558633 ["Character/Dirichlet/161/151"] "1-161-161.152-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.152" [[0, 0.0]] [] 0 true true false false 0.09069197200590509 0 1.81042723996 ["Character/Dirichlet/161/152"] "1-161-161.153-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.153" [[0, 0.0]] [] 0 true true false false -0.0703631340460373 0 1.36487910229 ["Character/Dirichlet/161/153"] "1-161-161.156-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.156" [[0, 0.0]] [] 0 true true false false -0.01654472859661724 0 1.97250495826 ["Character/Dirichlet/161/156"] "1-161-161.157-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.157" [[0, 0.0]] [] 0 true true false false -0.09134936471159363 0 1.36237630654 ["Character/Dirichlet/161/157"] "1-161-161.159-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.159" [[0, 0.0]] [] 0 true true false false -0.3033318135491557 0 0.872044548076 ["Character/Dirichlet/161/159"] "1-161-161.16-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.16" [[0, 0.0]] [] 0 true true false false 0.24936355918994513 0 1.77773628653 ["Character/Dirichlet/161/16"] "1-161-161.160-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.160" [[0, 0.0]] [] 0 true true true true 0.0 0 1.65723590716 ["Character/Dirichlet/161/160"] "1-161-161.17-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.17" [[0, 0.0]] [] 0 true true false false 0.13169527081948587 0 2.32180887835 ["Character/Dirichlet/161/17"] "1-161-161.18-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.18" [[0, 0.0]] [] 0 true true false false -0.20338434831723004 0 0.462241075132 ["Character/Dirichlet/161/18"] "1-161-161.19-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.19" [[0, 0.0]] [] 0 true true false false -0.13169527081948587 0 1.58594145598 ["Character/Dirichlet/161/19"] "1-161-161.2-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.2" [[0, 0.0]] [] 0 true true false false -0.4455914718382867 0 1.1520619415 ["Character/Dirichlet/161/2"] "1-161-161.20-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.20" [[0, 0.0]] [] 0 true true false false 0.0703631340460373 0 1.3083013056 ["Character/Dirichlet/161/20"] "1-161-161.25-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.25" [[0, 0.0]] [] 0 true true false false 0.36756890844110063 0 2.18105309592 ["Character/Dirichlet/161/25"] "1-161-161.32-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.32" [[0, 0.0]] [] 0 true true false false 0.01654472859661724 0 1.26526814947 ["Character/Dirichlet/161/32"] "1-161-161.33-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.33" [[0, 0.0]] [] 0 true true false false -0.038921362878720306 0 2.12754855492 ["Character/Dirichlet/161/33"] "1-161-161.34-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.34" [[0, 0.0]] [] 0 true true false false -0.2168411395047904 0 1.6171431162 ["Character/Dirichlet/161/34"] "1-161-161.38-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.38" [[0, 0.0]] [] 0 true true false false 0.44980981900790884 0 0.416358963728 ["Character/Dirichlet/161/38"] "1-161-161.39-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.39" [[0, 0.0]] [] 0 true true false false 0.06379682108944211 0 1.44486781501 ["Character/Dirichlet/161/39"] "1-161-161.4-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.4" [[0, 0.0]] [] 0 true true false false -0.05195358841774542 0 1.10993500402 ["Character/Dirichlet/161/4"] "1-161-161.40-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.40" [[0, 0.0]] [] 0 true true false false 0.09134936471159363 0 1.28288459268 ["Character/Dirichlet/161/40"] "1-161-161.45-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.45" [[0, 0.0]] [] 0 true true false false -0.4798270469460539 0 2.3881162192 ["Character/Dirichlet/161/45"] "1-161-161.5-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.5" [[0, 0.0]] [] 0 true true false false -0.13103787811379736 0 1.2247698481 ["Character/Dirichlet/161/5"] "1-161-161.58-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.58" [[0, 0.0]] [] 0 true true false false -0.36756890844110063 0 0.564654304433 ["Character/Dirichlet/161/58"] "1-161-161.61-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.61" [[0, 0.0]] [] 0 true true false false 0.4094639129000166 0 2.35994995101 ["Character/Dirichlet/161/61"] "1-161-161.66-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.66" [[0, 0.0]] [] 0 true true false false -0.4094639129000166 0 1.09146675535 ["Character/Dirichlet/161/66"] "1-161-161.68-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.68" [[0, 0.0]] [] 0 true true false false 0.4798270469460539 0 0.668678485833 ["Character/Dirichlet/161/68"] "1-161-161.72-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.72" [[0, 0.0]] [] 0 true true false false 0.22393030518028365 0 2.34535956729 ["Character/Dirichlet/161/72"] "1-161-161.76-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.76" [[0, 0.0]] [] 0 true true false false -0.4812515901752258 0 0.0993422392882 ["Character/Dirichlet/161/76"] "1-161-161.80-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.80" [[0, 0.0]] [] 0 true true false false 0.3033318135491557 0 2.17459984053 ["Character/Dirichlet/161/80"] "1-161-161.81-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.81" [[0, 0.0]] [] 0 true true false false 0.4455914718382867 0 3.06700790673 ["Character/Dirichlet/161/81"] "1-161-161.83-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.83" [[0, 0.0]] [] 0 true true false false -0.3891350749401488 0 0.969494185478 ["Character/Dirichlet/161/83"] "1-161-161.89-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.89" [[0, 0.0]] [] 0 true true false false -0.44980981900790884 0 2.78015544158 ["Character/Dirichlet/161/89"] "1-161-161.9-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.9" [[0, 0.0]] [] 0 true true false false 0.20338434831723004 0 1.76890913333 ["Character/Dirichlet/161/9"] "1-161-161.90-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.90" [[0, 0.0]] [] 0 true true false false 0.2168411395047904 0 2.19975760093 ["Character/Dirichlet/161/90"] "1-161-161.95-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.95" [[0, 0.0]] [] 0 true true false false 0.03709068545967084 0 1.7433873218 ["Character/Dirichlet/161/95"] "1-161-161.97-r0-0-0" 0.7476808566838746 0.7476808566838746 1 161 "161.97" [[0, 0.0]] [] 0 true true false false 0.3891350749401488 0 2.75789175028 ["Character/Dirichlet/161/97"] # 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).