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Properties

Label 2-31e2-31.19-c1-0-57
Degree $2$
Conductor $961$
Sign $-0.418 - 0.908i$
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  + (−1.02 − 0.744i)2-s + (−1.35 − 0.603i)3-s + (−0.122 − 0.376i)4-s + (1.90 − 3.29i)5-s + (0.940 + 1.62i)6-s + (1.46 − 1.62i)7-s + (−0.937 + 2.88i)8-s + (−0.533 − 0.592i)9-s + (−4.39 + 1.95i)10-s + (−0.929 − 0.197i)11-s + (−0.0613 + 0.584i)12-s + (0.0175 + 0.167i)13-s + (−2.71 + 0.576i)14-s + (−4.56 + 3.31i)15-s + (2.46 − 1.79i)16-s + (−6.43 + 1.36i)17-s + ⋯
L(s)  = 1  + (−0.724 − 0.526i)2-s + (−0.782 − 0.348i)3-s + (−0.0611 − 0.188i)4-s + (0.849 − 1.47i)5-s + (0.383 + 0.664i)6-s + (0.553 − 0.614i)7-s + (−0.331 + 1.02i)8-s + (−0.177 − 0.197i)9-s + (−1.39 + 0.619i)10-s + (−0.280 − 0.0595i)11-s + (−0.0177 + 0.168i)12-s + (0.00487 + 0.0463i)13-s + (−0.724 + 0.154i)14-s + (−1.17 + 0.856i)15-s + (0.617 − 0.448i)16-s + (−1.56 + 0.331i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.418 - 0.908i$
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: \((2,\ 961,\ (\ :1/2),\ -0.418 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.284414 + 0.444056i\)
\(L(\frac12)\) \(\approx\) \(0.284414 + 0.444056i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.02 + 0.744i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (1.35 + 0.603i)T + (2.00 + 2.22i)T^{2} \)
5 \( 1 + (-1.90 + 3.29i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.46 + 1.62i)T + (-0.731 - 6.96i)T^{2} \)
11 \( 1 + (0.929 + 0.197i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (-0.0175 - 0.167i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (6.43 - 1.36i)T + (15.5 - 6.91i)T^{2} \)
19 \( 1 + (-0.120 + 1.14i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (-1.43 + 4.40i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-1.08 - 0.785i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (1.93 + 3.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.299 + 0.133i)T + (27.4 - 30.4i)T^{2} \)
43 \( 1 + (-1.00 + 9.57i)T + (-42.0 - 8.94i)T^{2} \)
47 \( 1 + (4.56 - 3.31i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-4.90 - 5.44i)T + (-5.54 + 52.7i)T^{2} \)
59 \( 1 + (2.42 + 1.07i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 - 1.74T + 61T^{2} \)
67 \( 1 + (0.276 - 0.478i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.760 - 0.844i)T + (-7.42 + 70.6i)T^{2} \)
73 \( 1 + (7.74 + 1.64i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (-4.44 + 0.944i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (0.298 - 0.132i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (-4.54 - 13.9i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-4.79 - 14.7i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.270668427917456273606031071621, −8.910688889736165981822451728675, −8.161112388640195900697741032180, −6.78251507775457179367180647015, −5.86616944578993897379910382975, −5.12676785352398802607071224732, −4.41685871617348188205767081671, −2.28633066734431371833677633384, −1.27753564718423573256502125859, −0.36510858813843074723167484179, 2.15348967293802311576031020395, 3.18409150264322877628392313302, 4.67358055381733514108467750347, 5.69943340511146620100669128405, 6.42799398314418695766419445065, 7.11984225336963655773309221855, 8.077167038503805113081613705551, 8.944974513432220833154137624411, 9.835772851114916669370412647135, 10.43393949036625728999525085153

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