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Properties

Label 2-31e2-31.3-c0-0-0
Degree $2$
Conductor $961$
Sign $0.669 + 0.742i$
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 + (0.978 − 0.207i)7-s + (0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (−0.104 − 0.994i)10-s + (0.669 − 0.743i)14-s + (0.809 + 0.587i)16-s + (−0.913 + 0.406i)18-s + (−0.913 − 0.406i)19-s + (0.309 − 0.951i)35-s + (−0.978 + 0.207i)38-s + (0.978 + 0.207i)40-s + (0.104 + 0.994i)41-s + (−0.669 + 0.743i)45-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.5 − 0.866i)5-s + (0.978 − 0.207i)7-s + (0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (−0.104 − 0.994i)10-s + (0.669 − 0.743i)14-s + (0.809 + 0.587i)16-s + (−0.913 + 0.406i)18-s + (−0.913 − 0.406i)19-s + (0.309 − 0.951i)35-s + (−0.978 + 0.207i)38-s + (0.978 + 0.207i)40-s + (0.104 + 0.994i)41-s + (−0.669 + 0.743i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.669 + 0.742i$
Analytic conductor: \(0.479601\)
Root analytic conductor: \(0.692532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (623, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :0),\ 0.669 + 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.626151421\)
\(L(\frac12)\) \(\approx\) \(1.626151421\)
\(L(1)\) 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 \( 1 + (0.978 + 0.207i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2} \)
11 \( 1 + (0.104 - 0.994i)T^{2} \)
13 \( 1 + (-0.669 + 0.743i)T^{2} \)
17 \( 1 + (0.104 + 0.994i)T^{2} \)
19 \( 1 + (0.913 + 0.406i)T + (0.669 + 0.743i)T^{2} \)
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 \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \)
43 \( 1 + (-0.669 - 0.743i)T^{2} \)
47 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.913 - 0.406i)T^{2} \)
59 \( 1 + (-0.104 + 0.994i)T + (-0.978 - 0.207i)T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \)
73 \( 1 + (0.104 - 0.994i)T^{2} \)
79 \( 1 + (0.104 + 0.994i)T^{2} \)
83 \( 1 + (0.978 - 0.207i)T^{2} \)
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

−10.33478660072070114911709384284, −9.142718481717725260669551767750, −8.469154606519973285836416536759, −7.896979140231161597035985485921, −6.46101099466718733896955973402, −5.32493451157343929425099773652, −4.89654453698213806799888124776, −3.91734267389053394036883341751, −2.71665754552043616045680626578, −1.62630539029025569950701040810, 1.91889955753860796018432583885, 3.12073496653533552091809481045, 4.37816571753116951183572140188, 5.24487836005615928883069115786, 6.01542565225747719005281052547, 6.62550722293659408774453356583, 7.69554190136341208031224248106, 8.509432000786197797439662747301, 9.549075258622067492864488500434, 10.54791896417492195793180778812

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