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Properties

Label 2-31e2-31.4-c1-0-8
Degree $2$
Conductor $961$
Sign $0.800 - 0.598i$
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  + (−0.831 − 2.55i)2-s + (−0.438 + 1.34i)3-s + (−4.23 + 3.07i)4-s + 0.608·5-s + 3.81·6-s + (1.39 − 1.01i)7-s + (7.04 + 5.11i)8-s + (0.797 + 0.579i)9-s + (−0.505 − 1.55i)10-s + (−1.08 + 0.786i)11-s + (−2.29 − 7.06i)12-s + (−1.13 + 3.49i)13-s + (−3.75 − 2.73i)14-s + (−0.266 + 0.821i)15-s + (4.00 − 12.3i)16-s + (−2.23 − 1.62i)17-s + ⋯
L(s)  = 1  + (−0.587 − 1.80i)2-s + (−0.253 + 0.779i)3-s + (−2.11 + 1.53i)4-s + 0.272·5-s + 1.55·6-s + (0.528 − 0.383i)7-s + (2.49 + 1.80i)8-s + (0.265 + 0.193i)9-s + (−0.159 − 0.492i)10-s + (−0.326 + 0.237i)11-s + (−0.662 − 2.04i)12-s + (−0.314 + 0.968i)13-s + (−1.00 − 0.729i)14-s + (−0.0688 + 0.212i)15-s + (1.00 − 3.07i)16-s + (−0.542 − 0.393i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.800 - 0.598i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ 0.800 - 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.576383 + 0.191660i\)
\(L(\frac12)\) \(\approx\) \(0.576383 + 0.191660i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.831 + 2.55i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.438 - 1.34i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 - 0.608T + 5T^{2} \)
7 \( 1 + (-1.39 + 1.01i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (1.08 - 0.786i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.13 - 3.49i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.23 + 1.62i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.793 + 2.44i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.436 + 0.316i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.51 - 7.73i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 7.74T + 37T^{2} \)
41 \( 1 + (0.0321 + 0.0989i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-0.928 - 2.85i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (2.07 - 6.39i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.26 - 1.64i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.144 + 0.443i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 5.11T + 61T^{2} \)
67 \( 1 - 8.29T + 67T^{2} \)
71 \( 1 + (3.84 + 2.79i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-6.07 + 4.41i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (7.84 + 5.70i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.15 - 15.8i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (12.3 - 9.00i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.03 + 0.751i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.24645280779258647757605689075, −9.524938361847310752097256700105, −8.984626496918636215218618083117, −7.967203838685021605738112797786, −6.97791119586438202762520710493, −5.08229912329363817900728936408, −4.56691135474800104502125426534, −3.74603088791178889816227499443, −2.44890481302600708722926292523, −1.50182692631369926074183869676, 0.37094832384168314283848817885, 1.87628297791212400545230891004, 4.07273756478420922443209136029, 5.29312370057289781271167632024, 5.83159859184859409735701574875, 6.61291218808040080105983341379, 7.40739360159239509346537799823, 8.149820638111946379432252747350, 8.603316243545502670490176295038, 9.835481550957362046950261609917

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