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Properties

Label 2-31e2-31.16-c1-0-8
Degree $2$
Conductor $961$
Sign $-0.762 - 0.647i$
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  + (0.5 − 0.363i)2-s + (−1 − 0.726i)3-s + (−0.5 + 1.53i)4-s + 5-s − 0.763·6-s + (−1.30 + 4.02i)7-s + (0.690 + 2.12i)8-s + (−0.454 − 1.40i)9-s + (0.5 − 0.363i)10-s + (0.618 − 1.90i)11-s + (1.61 − 1.17i)12-s + (−1 − 0.726i)13-s + (0.809 + 2.48i)14-s + (−1 − 0.726i)15-s + (−1.49 − 1.08i)16-s + (1.61 + 4.97i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.256i)2-s + (−0.577 − 0.419i)3-s + (−0.250 + 0.769i)4-s + 0.447·5-s − 0.311·6-s + (−0.494 + 1.52i)7-s + (0.244 + 0.751i)8-s + (−0.151 − 0.466i)9-s + (0.158 − 0.114i)10-s + (0.186 − 0.573i)11-s + (0.467 − 0.339i)12-s + (−0.277 − 0.201i)13-s + (0.216 + 0.665i)14-s + (−0.258 − 0.187i)15-s + (−0.374 − 0.272i)16-s + (0.392 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.762 - 0.647i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ -0.762 - 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.219097 + 0.596707i\)
\(L(\frac12)\) \(\approx\) \(0.219097 + 0.596707i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.363i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (1 + 0.726i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 + (1.30 - 4.02i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (-0.618 + 1.90i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (1 + 0.726i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.61 - 4.97i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.80 - 1.31i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.38 + 7.33i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (5.85 - 4.25i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (5.66 - 4.11i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-2.61 + 1.90i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-5.23 - 3.80i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.472 + 1.45i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.80 - 1.31i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + (-4.07 - 12.5i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.145 - 0.449i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-0.527 - 1.62i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.38 - 1.73i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.527 - 1.62i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-0.600 + 1.84i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.47668949226914904547087242294, −9.345723575662514500416060803066, −8.700158147465288834147326943585, −8.017325510335979143650473428269, −6.66641465080018694362987123256, −5.93079969477016281788853714201, −5.40698026196953901136081411942, −3.96890152504501207392750691857, −3.01305391650050421367014900410, −1.96692944167572590760555932771, 0.26848460114745219926570147508, 1.85899991109499497841123718049, 3.72088972374755990463734373785, 4.51702101260879024026853345430, 5.31965735447978908765507505358, 6.08057523351771029454983859615, 7.09564495753994854094487202708, 7.63747526276266042585125793829, 9.415633351676881190578847180081, 9.742494216546238320127615463036

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