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Properties

Label 2-31e2-31.4-c1-0-1
Degree $2$
Conductor $961$
Sign $0.999 + 0.0296i$
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  + (−0.746 − 2.29i)2-s + (0.746 − 2.29i)3-s + (−3.09 + 2.25i)4-s + 5-s − 5.82·6-s + (−1.95 + 1.41i)7-s + (3.57 + 2.59i)8-s + (−2.28 − 1.66i)9-s + (−0.746 − 2.29i)10-s + (−4.24 + 3.08i)11-s + (2.85 + 8.79i)12-s + (−0.565 + 1.73i)13-s + (4.71 + 3.42i)14-s + (0.746 − 2.29i)15-s + (0.927 − 2.85i)16-s + (−0.138 − 0.100i)17-s + ⋯
L(s)  = 1  + (−0.527 − 1.62i)2-s + (0.430 − 1.32i)3-s + (−1.54 + 1.12i)4-s + 0.447·5-s − 2.37·6-s + (−0.738 + 0.536i)7-s + (1.26 + 0.917i)8-s + (−0.762 − 0.554i)9-s + (−0.235 − 0.726i)10-s + (−1.27 + 0.929i)11-s + (0.824 + 2.53i)12-s + (−0.156 + 0.482i)13-s + (1.26 + 0.915i)14-s + (0.192 − 0.592i)15-s + (0.231 − 0.713i)16-s + (−0.0336 − 0.0244i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.999 + 0.0296i$
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: \((2,\ 961,\ (\ :1/2),\ 0.999 + 0.0296i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.256756 - 0.00380512i\)
\(L(\frac12)\) \(\approx\) \(0.256756 - 0.00380512i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.746 + 2.29i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.746 + 2.29i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 + (1.95 - 1.41i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (4.24 - 3.08i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.565 - 1.73i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.138 + 0.100i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.490 - 1.50i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.23 + 2.35i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.362 - 1.11i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-2.93 - 9.02i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (2.75 + 8.46i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (0.511 - 1.57i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.138 - 0.100i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.11 - 9.57i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 2.82T + 61T^{2} \)
67 \( 1 + 5.24T + 67T^{2} \)
71 \( 1 + (-11.3 - 8.27i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.09 + 2.25i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (12.3 + 8.95i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.25 + 3.87i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (10.1 - 7.33i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (8.76 - 6.36i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.956416828527864287533653865730, −9.466810580449735371250904227570, −8.475282457006405801629168186986, −7.79666650643425049998252879498, −6.82891367774847080479219380476, −5.77166632157364817100277335783, −4.36466186639473103709760481876, −2.93664624550242929592999052010, −2.33920794356340236606440686716, −1.58562171812125841949779020464, 0.12872691484090804017722086150, 2.91045123429347156003971823042, 3.96343562414858544794805002969, 5.13420633684413546295092471172, 5.68131302402784428476655008613, 6.61174861870391128685084160970, 7.70412079685093600667913423973, 8.281722112673303899751569282688, 9.193595415850191604720973124334, 9.837041555468624982542413571152

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