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Properties

Label 2-31e2-31.12-c0-0-0
Degree $2$
Conductor $961$
Sign $0.987 - 0.157i$
Analytic cond. $0.479601$
Root an. cond. $0.692532$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)2-s + (0.5 − 0.866i)5-s + (−0.669 + 0.743i)7-s + (0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.913 − 0.406i)10-s + (−0.978 + 0.207i)14-s + (0.809 − 0.587i)16-s + (0.104 + 0.994i)18-s + (0.104 − 0.994i)19-s + (0.309 + 0.951i)35-s + (0.669 − 0.743i)38-s + (−0.669 − 0.743i)40-s + (−0.913 + 0.406i)41-s + (0.978 − 0.207i)45-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (0.5 − 0.866i)5-s + (−0.669 + 0.743i)7-s + (0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (0.913 − 0.406i)10-s + (−0.978 + 0.207i)14-s + (0.809 − 0.587i)16-s + (0.104 + 0.994i)18-s + (0.104 − 0.994i)19-s + (0.309 + 0.951i)35-s + (0.669 − 0.743i)38-s + (−0.669 − 0.743i)40-s + (−0.913 + 0.406i)41-s + (0.978 − 0.207i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.987 - 0.157i$
Analytic conductor: \(0.479601\)
Root analytic conductor: \(0.692532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :0),\ 0.987 - 0.157i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.565690949\)
\(L(\frac12)\) \(\approx\) \(1.565690949\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (-0.669 - 0.743i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2} \)
11 \( 1 + (-0.913 - 0.406i)T^{2} \)
13 \( 1 + (0.978 - 0.207i)T^{2} \)
17 \( 1 + (-0.913 + 0.406i)T^{2} \)
19 \( 1 + (-0.104 + 0.994i)T + (-0.978 - 0.207i)T^{2} \)
23 \( 1 + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \)
43 \( 1 + (0.978 + 0.207i)T^{2} \)
47 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.104 - 0.994i)T^{2} \)
59 \( 1 + (0.913 + 0.406i)T + (0.669 + 0.743i)T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2} \)
73 \( 1 + (-0.913 - 0.406i)T^{2} \)
79 \( 1 + (-0.913 + 0.406i)T^{2} \)
83 \( 1 + (-0.669 + 0.743i)T^{2} \)
89 \( 1 + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.999292838019451830328868149768, −9.470019250949657410065883717322, −8.666034385778887218707763119738, −7.50874444286084597827315918927, −6.61160531332545103609261591002, −5.81994200762470407767016058452, −5.01593406507155790467970971622, −4.47662477328812333837678216858, −3.03803085588313957998326853375, −1.54376595017761818053877068744, 1.82322750323479001888303308449, 3.17670698977402126135125818953, 3.66391454847259638094659734680, 4.67262956409460185881218287176, 5.92609107125770357041190415887, 6.70843601474910535442376309376, 7.47277656510777758387005073915, 8.576769874602581919412023267016, 9.807526051751717952296328020812, 10.22520036209694947744195062403

Graph of the $Z$-function along the critical line