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Properties

Label 2-31e2-31.19-c1-0-28
Degree $2$
Conductor $961$
Sign $0.949 - 0.314i$
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  + (−1.30 − 0.951i)2-s + (2.95 + 1.31i)3-s + (0.190 + 0.587i)4-s + (−0.5 + 0.866i)5-s + (−2.61 − 4.53i)6-s + (0.157 − 0.175i)7-s + (−0.690 + 2.12i)8-s + (4.99 + 5.55i)9-s + (1.47 − 0.658i)10-s + (1.95 + 0.415i)11-s + (−0.209 + 1.98i)12-s + (−0.338 − 3.21i)13-s + (−0.373 + 0.0794i)14-s + (−2.61 + 1.90i)15-s + (3.92 − 2.85i)16-s + (0.747 − 0.158i)17-s + ⋯
L(s)  = 1  + (−0.925 − 0.672i)2-s + (1.70 + 0.759i)3-s + (0.0954 + 0.293i)4-s + (−0.223 + 0.387i)5-s + (−1.06 − 1.85i)6-s + (0.0597 − 0.0663i)7-s + (−0.244 + 0.751i)8-s + (1.66 + 1.85i)9-s + (0.467 − 0.208i)10-s + (0.589 + 0.125i)11-s + (−0.0603 + 0.574i)12-s + (−0.0938 − 0.892i)13-s + (−0.0998 + 0.0212i)14-s + (−0.675 + 0.491i)15-s + (0.981 − 0.713i)16-s + (0.181 − 0.0385i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.949 - 0.314i$
Analytic conductor: \(7.67362\)
Root analytic conductor: \(2.77013\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (732, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :1/2),\ 0.949 - 0.314i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70496 + 0.275511i\)
\(L(\frac12)\) \(\approx\) \(1.70496 + 0.275511i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.30 + 0.951i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-2.95 - 1.31i)T + (2.00 + 2.22i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.157 + 0.175i)T + (-0.731 - 6.96i)T^{2} \)
11 \( 1 + (-1.95 - 0.415i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (0.338 + 3.21i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (-0.747 + 0.158i)T + (15.5 - 6.91i)T^{2} \)
19 \( 1 + (-0.233 + 2.22i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (1.76 - 5.42i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.23 - 1.62i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.39 + 2.84i)T + (27.4 - 30.4i)T^{2} \)
43 \( 1 + (-0.129 + 1.22i)T + (-42.0 - 8.94i)T^{2} \)
47 \( 1 + (2 - 1.45i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-7.00 - 7.78i)T + (-5.54 + 52.7i)T^{2} \)
59 \( 1 + (-2.04 - 0.909i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + 8.18T + 61T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.14 + 6.82i)T + (-7.42 + 70.6i)T^{2} \)
73 \( 1 + (-8.28 - 1.76i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (11.4 - 2.43i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (-13.6 + 6.07i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (3.61 + 11.1i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.92 + 15.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.946329193947377067355622730604, −9.179168606874644148307493889177, −8.809636851381838742343708043772, −7.80263293308365736895663484499, −7.33324318063674385476533728681, −5.58083511073306263934370106308, −4.39582506513835105745450693415, −3.31145309853263925104729576062, −2.69264545887661440283173261105, −1.47557155684217227106171692713, 1.02874398411419716296613537054, 2.29607635489839945317745321525, 3.56023356987468585740590288174, 4.35652633187241642517758580828, 6.36642021827019285491018423839, 6.84453833934405824115446081525, 7.80143734952898177262368619920, 8.319886547756006381700471053969, 8.860709610671396030333403916115, 9.484813365600693533121657082539

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