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Properties

Label 2-31e2-31.2-c1-0-4
Degree $2$
Conductor $961$
Sign $0.0229 - 0.999i$
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  + (−1.67 − 1.21i)2-s + (−1.73 + 1.25i)3-s + (0.708 + 2.18i)4-s + 2.34·5-s + 4.43·6-s + (−1.13 − 3.49i)7-s + (0.188 − 0.578i)8-s + (0.487 − 1.49i)9-s + (−3.93 − 2.85i)10-s + (1.32 + 4.07i)11-s + (−3.96 − 2.88i)12-s + (−1.68 + 1.22i)13-s + (−2.35 + 7.24i)14-s + (−4.06 + 2.95i)15-s + (2.69 − 1.95i)16-s + (−0.657 + 2.02i)17-s + ⋯
L(s)  = 1  + (−1.18 − 0.861i)2-s + (−0.999 + 0.726i)3-s + (0.354 + 1.09i)4-s + 1.05·5-s + 1.80·6-s + (−0.429 − 1.32i)7-s + (0.0664 − 0.204i)8-s + (0.162 − 0.499i)9-s + (−1.24 − 0.904i)10-s + (0.398 + 1.22i)11-s + (−1.14 − 0.832i)12-s + (−0.468 + 0.340i)13-s + (−0.628 + 1.93i)14-s + (−1.04 + 0.762i)15-s + (0.672 − 0.488i)16-s + (−0.159 + 0.490i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0229 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0229 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.0229 - 0.999i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ 0.0229 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.237044 + 0.231673i\)
\(L(\frac12)\) \(\approx\) \(0.237044 + 0.231673i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 + 1.21i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (1.73 - 1.25i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 - 2.34T + 5T^{2} \)
7 \( 1 + (1.13 + 3.49i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (-1.32 - 4.07i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.68 - 1.22i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.657 - 2.02i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.501 - 0.364i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.03 + 3.18i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.11 + 0.809i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + 0.274T + 37T^{2} \)
41 \( 1 + (-3.46 - 2.51i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-0.218 - 0.158i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (4.35 - 3.16i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.93 - 9.04i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.71 + 3.42i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + 2.22T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 + (0.412 - 1.27i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.37 - 13.4i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.68 - 8.25i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.19 + 3.04i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.54 + 4.76i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.07 - 6.38i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.15345138437459506195346844558, −9.773452924794486549702458985968, −9.153343733700126269715252222557, −7.81736247778935856413662107262, −6.89400360604140915797551400417, −5.98875928338815776927298775310, −4.85124971741426048753195356643, −4.01683751962004818631365882604, −2.42129790789967357495447007923, −1.27855203408676188065079493490, 0.28184314643316443956733133670, 1.65507603775574289958192799563, 3.10522240786527804944500560874, 5.41736613749588517613897183618, 5.78199468414452014363958108263, 6.39532532403027307608307274337, 7.12739213385407893359462444693, 8.184937165838825372723781748686, 9.143623021172308174576334034705, 9.398646785558113940299634622473

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