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Properties

Label 2-31e2-31.8-c1-0-5
Degree $2$
Conductor $961$
Sign $0.0525 - 0.998i$
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  + (0.713 − 2.19i)2-s + (0.785 + 2.41i)3-s + (−2.68 − 1.95i)4-s − 2.49·5-s + 5.86·6-s + (−1.29 − 0.941i)7-s + (−2.47 + 1.79i)8-s + (−2.80 + 2.03i)9-s + (−1.78 + 5.48i)10-s + (0.593 + 0.431i)11-s + (2.61 − 8.03i)12-s + (0.585 + 1.80i)13-s + (−2.99 + 2.17i)14-s + (−1.96 − 6.04i)15-s + (0.124 + 0.384i)16-s + (−3.54 + 2.57i)17-s + ⋯
L(s)  = 1  + (0.504 − 1.55i)2-s + (0.453 + 1.39i)3-s + (−1.34 − 0.977i)4-s − 1.11·5-s + 2.39·6-s + (−0.489 − 0.355i)7-s + (−0.874 + 0.635i)8-s + (−0.934 + 0.678i)9-s + (−0.563 + 1.73i)10-s + (0.178 + 0.130i)11-s + (0.754 − 2.32i)12-s + (0.162 + 0.500i)13-s + (−0.799 + 0.580i)14-s + (−0.506 − 1.56i)15-s + (0.0312 + 0.0960i)16-s + (−0.859 + 0.624i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0525 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0525 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.0525 - 0.998i$
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: \((2,\ 961,\ (\ :1/2),\ 0.0525 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.459246 + 0.435724i\)
\(L(\frac12)\) \(\approx\) \(0.459246 + 0.435724i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.713 + 2.19i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.785 - 2.41i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + 2.49T + 5T^{2} \)
7 \( 1 + (1.29 + 0.941i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-0.593 - 0.431i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.585 - 1.80i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (3.54 - 2.57i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.43 - 4.41i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (5.75 - 4.18i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.0398 + 0.122i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 8.42T + 37T^{2} \)
41 \( 1 + (2.27 - 7.01i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (0.0712 - 0.219i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (2.48 + 7.64i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.63 + 3.37i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.93 - 9.03i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 7.84T + 61T^{2} \)
67 \( 1 - 4.82T + 67T^{2} \)
71 \( 1 + (-2.75 + 2.00i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.17 - 1.58i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.66 + 2.66i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.825 - 2.53i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (1.78 + 1.29i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (9.94 + 7.22i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.13891932582356319978436239070, −9.928074349744737010386740928740, −8.882865385600584492259733813381, −8.139921516745580233114424646791, −6.79614673764623115066306538576, −5.32006687036229781196569747484, −4.16057375466177920836534053398, −3.94026901159330246134798621907, −3.34014365016814768715346224078, −1.91208460628878764704604611697, 0.22532914477825193375949485169, 2.38123932808696742197914133882, 3.66360441629272783302313001923, 4.68987384876542656475497323637, 5.84773245369507318811698496084, 6.79978139675404916506529760258, 7.00459827942929016065225042328, 8.060608385180809573751965049520, 8.355717273555426557634110307481, 9.214278659631135212738466375703

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