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Properties

Label 2-31e2-31.4-c1-0-35
Degree $2$
Conductor $961$
Sign $-0.634 + 0.773i$
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  + (−0.309 − 0.951i)2-s + (−0.874 + 2.68i)3-s + (0.809 − 0.587i)4-s + 2.82·6-s + (−3.23 + 2.35i)7-s + (−2.42 − 1.76i)8-s + (−4.04 − 2.93i)9-s + (2.28 − 1.66i)11-s + (0.874 + 2.68i)12-s + (−0.437 + 1.34i)13-s + (3.23 + 2.35i)14-s + (−0.309 + 0.951i)16-s + (1.14 + 0.831i)17-s + (−1.54 + 4.75i)18-s + (−1.23 − 3.80i)19-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.504 + 1.55i)3-s + (0.404 − 0.293i)4-s + 1.15·6-s + (−1.22 + 0.888i)7-s + (−0.858 − 0.623i)8-s + (−1.34 − 0.979i)9-s + (0.689 − 0.501i)11-s + (0.252 + 0.776i)12-s + (−0.121 + 0.373i)13-s + (0.864 + 0.628i)14-s + (−0.0772 + 0.237i)16-s + (0.277 + 0.201i)17-s + (−0.364 + 1.12i)18-s + (−0.283 − 0.872i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.634 + 0.773i$
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: \((2,\ 961,\ (\ :1/2),\ -0.634 + 0.773i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.127006 - 0.268505i\)
\(L(\frac12)\) \(\approx\) \(0.127006 - 0.268505i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.874 - 2.68i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 + (3.23 - 2.35i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-2.28 + 1.66i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.437 - 1.34i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.14 - 0.831i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.23 + 3.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (4.57 + 3.32i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.437 - 1.34i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 4.24T + 37T^{2} \)
41 \( 1 + (0.618 + 1.90i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (2.62 + 8.06i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-3.70 + 11.4i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.43 - 2.49i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.47 + 7.60i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 1.41T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (6.47 + 4.70i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (3.43 - 2.49i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (9.15 + 6.65i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.37 - 13.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (5.72 - 4.15i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (6.47 - 4.70i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.912160220449787086122633418382, −9.217543311789550128446599770618, −8.695868073426479530663449270290, −6.81262061253140831896256908294, −6.07255889609660993157711140221, −5.47987596220005335123217619709, −4.10264067400641663459788834061, −3.38012084922844866881492975228, −2.31867655238823765203183921213, −0.15113960954661166720935292978, 1.46741319083444709556948620869, 2.80101347493079498441657035596, 4.00613070358330865625264534731, 5.94297925892233716733500501464, 6.12034666130550408409236376899, 7.04275041098555200846254583736, 7.55024240567844436568302233407, 8.127330590097757818572508212239, 9.455200924240857086833541392578, 10.22918260959923321436381468597

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