# lfunc_search downloaded from the LMFDB on 30 April 2026. # Search link: https://www.lmfdb.org/L/1/152/152.139 # Query "{'degree': 1, 'conductor': 152}" returned 34 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. "1-152-152.101-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.101" [[0, 0.0]] [] 0 true true false false -0.2801894400283459 0 0.835957045193 ["Character/Dirichlet/152/101"] "1-152-152.107-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.107" [[0, 0.0]] [] 0 true true false false -0.26558004527105655 0 1.06790722964 ["Character/Dirichlet/152/107"] "1-152-152.125-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.125" [[0, 0.0]] [] 0 true true false false 0.20054331069780507 0 2.20318121969 ["Character/Dirichlet/152/125"] "1-152-152.147-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.147" [[0, 0.0]] [] 0 true true false false -0.09194774279626558 0 1.34125820926 ["Character/Dirichlet/152/147"] "1-152-152.149-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.149" [[0, 0.0]] [] 0 true true false false 0.2801894400283459 0 1.97495632397 ["Character/Dirichlet/152/149"] "1-152-152.27-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.27" [[0, 0.0]] [] 0 true true false false 0.26558004527105655 0 2.35764206689 ["Character/Dirichlet/152/27"] "1-152-152.3-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.3" [[0, 0.0]] [] 0 true true false false 0.055659436514559055 0 1.35826066962 ["Character/Dirichlet/152/3"] "1-152-152.45-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.45" [[0, 0.0]] [] 0 true true false false -0.20054331069780507 0 1.37602865796 ["Character/Dirichlet/152/45"] "1-152-152.5-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.5" [[0, 0.0]] [] 0 true true false false 0.0339805249901419 0 1.69084118533 ["Character/Dirichlet/152/5"] "1-152-152.51-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.51" [[0, 0.0]] [] 0 true true false false -0.055659436514559055 0 1.3938469463 ["Character/Dirichlet/152/51"] "1-152-152.59-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.59" [[0, 0.0]] [] 0 true true false false -0.3647983151139036 0 0.592353271556 ["Character/Dirichlet/152/59"] "1-152-152.61-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.61" [[0, 0.0]] [] 0 true true false false -0.0339805249901419 0 1.53281707828 ["Character/Dirichlet/152/61"] "1-152-152.67-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.67" [[0, 0.0]] [] 0 true true false false 0.3647983151139036 0 2.16428449108 ["Character/Dirichlet/152/67"] "1-152-152.75-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.75" [[0, 0.0]] [] 0 true true true true 0.0 0 0.982619117315 ["Character/Dirichlet/152/75"] "1-152-152.85-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.85" [[0, 0.0]] [] 0 true true false false -0.21970667901265062 0 0.534731972712 ["Character/Dirichlet/152/85"] "1-152-152.91-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.91" [[0, 0.0]] [] 0 true true false false 0.09194774279626558 0 2.11777660227 ["Character/Dirichlet/152/91"] "1-152-152.93-r0-0-0" 0.7058850323971985 0.7058850323971985 1 152 "152.93" [[0, 0.0]] [] 0 true true false false 0.21970667901265062 0 1.91681994033 ["Character/Dirichlet/152/93"] "1-152-152.109-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.109" [[1, 0.0]] [] 0 true true false false -0.09194774279626558 0 0.718949537412 ["Character/Dirichlet/152/109"] "1-152-152.11-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.11" [[1, 0.0]] [] 0 true true false false -0.29945668930219493 0 0.473583690225 ["Character/Dirichlet/152/11"] "1-152-152.117-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.117" [[1, 0.0]] [] 0 true true false false 0.055659436514559055 0 1.61630031234 ["Character/Dirichlet/152/117"] "1-152-152.123-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.123" [[1, 0.0]] [] 0 true true false false 0.28029332098734944 0 1.55105761304 ["Character/Dirichlet/152/123"] "1-152-152.13-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.13" [[1, 0.0]] [] 0 true true false false -0.055659436514559055 0 1.05925045675 ["Character/Dirichlet/152/13"] "1-152-152.131-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.131" [[1, 0.0]] [] 0 true true false false -0.28029332098734944 0 0.13712223537 ["Character/Dirichlet/152/131"] "1-152-152.139-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.139" [[1, 0.0]] [] 0 true true false false 0.2198105599716541 0 1.89702399863 ["Character/Dirichlet/152/139"] "1-152-152.141-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.141" [[1, 0.0]] [] 0 true true false false 0.26558004527105655 0 1.66345623241 ["Character/Dirichlet/152/141"] "1-152-152.21-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.21" [[1, 0.0]] [] 0 true true false false -0.3647983151139036 0 2.57720688632 ["Character/Dirichlet/152/21"] "1-152-152.29-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.29" [[1, 0.0]] [] 0 true true false false 0.3647983151139036 0 0.203111391144 ["Character/Dirichlet/152/29"] "1-152-152.35-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.35" [[1, 0.0]] [] 0 true true false false -0.2198105599716541 0 0.97644975791 ["Character/Dirichlet/152/35"] "1-152-152.37-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.37" [[1, 0.0]] [] 0 true true true true 0.0 0 1.19113351272 ["Character/Dirichlet/152/37"] "1-152-152.43-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.43" [[1, 0.0]] [] 0 true true false false -0.46601947500985813 0 0.439863178767 ["Character/Dirichlet/152/43"] "1-152-152.53-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.53" [[1, 0.0]] [] 0 true true false false 0.09194774279626558 0 0.826905032179 ["Character/Dirichlet/152/53"] "1-152-152.69-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.69" [[1, 0.0]] [] 0 true true false false -0.26558004527105655 0 0.382659961949 ["Character/Dirichlet/152/69"] "1-152-152.83-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.83" [[1, 0.0]] [] 0 true true false false 0.29945668930219493 0 1.36509272506 ["Character/Dirichlet/152/83"] "1-152-152.99-r1-0-0" 16.334668568783 16.334668568783 1 152 "152.99" [[1, 0.0]] [] 0 true true false false 0.46601947500985813 0 1.81089393402 ["Character/Dirichlet/152/99"] # 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).