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Properties

Label 2-31e2-31.8-c1-0-58
Degree $2$
Conductor $961$
Sign $-0.634 - 0.773i$
Analytic cond. $7.67362$
Root an. cond. $2.77013$
Motivic weight $1$
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 − 2.48i)2-s + (−0.270 − 0.831i)3-s + (−3.92 − 2.85i)4-s + 2.23·5-s − 2.28·6-s + (0.809 + 0.587i)7-s + (−6.04 + 4.39i)8-s + (1.80 − 1.31i)9-s + (1.80 − 5.56i)10-s + (−3.43 − 2.49i)11-s + (−1.31 + 4.03i)12-s + (−0.810 − 2.49i)13-s + (2.11 − 1.53i)14-s + (−0.603 − 1.85i)15-s + (3.04 + 9.37i)16-s + (−2.99 + 2.17i)17-s + ⋯
L(s)  = 1  + (0.572 − 1.76i)2-s + (−0.155 − 0.479i)3-s + (−1.96 − 1.42i)4-s + 0.999·5-s − 0.934·6-s + (0.305 + 0.222i)7-s + (−2.13 + 1.55i)8-s + (0.603 − 0.438i)9-s + (0.572 − 1.76i)10-s + (−1.03 − 0.751i)11-s + (−0.378 + 1.16i)12-s + (−0.224 − 0.691i)13-s + (0.566 − 0.411i)14-s + (−0.155 − 0.479i)15-s + (0.761 + 2.34i)16-s + (−0.726 + 0.527i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 - 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.634 - 0.773i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (628, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ -0.634 - 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.788511 + 1.66699i\)
\(L(\frac12)\) \(\approx\) \(0.788511 + 1.66699i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 2.48i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (0.270 + 0.831i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 - 2.23T + 5T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (3.43 + 2.49i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.810 + 2.49i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.99 - 2.17i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-2.12 + 1.54i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.166 - 0.513i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 + (0.454 - 1.40i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (2.99 - 9.21i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (3 + 9.23i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-11.1 + 8.06i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.69 - 11.3i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 13.9T + 61T^{2} \)
67 \( 1 - 6T + 67T^{2} \)
71 \( 1 + (-1.19 + 0.865i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.43 + 2.49i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.31 + 0.952i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.977 - 3.00i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-12.4 - 9.02i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-5.66 - 4.11i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.00504981845540124063279580986, −8.984281740249266255330574409110, −8.122937799561254786250447105542, −6.63479504511035236471447350797, −5.62329238537964813251701541826, −5.04514601238930271232873031387, −3.83552227282569033900427694556, −2.68938325104183145343888789853, −1.95146203857152594286689073823, −0.72472226765662158350568612095, 2.22461815705677917084077702266, 3.98927581358723747433060299358, 4.85196320307055914442872880742, 5.25321024806115769127921554392, 6.25353971627208944125090053695, 7.18405933684475966066091176758, 7.64185088487646394705999478159, 8.765350341427867159838428631273, 9.598467023757923049079035854740, 10.17365833385053135673537924721

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