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Properties

Label 2-31e2-31.19-c1-0-33
Degree $2$
Conductor $961$
Sign $-0.984 + 0.175i$
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  + (−1.67 − 1.21i)2-s + (1.95 + 0.870i)3-s + (0.708 + 2.18i)4-s + (−1.17 + 2.03i)5-s + (−2.21 − 3.83i)6-s + (−2.45 + 2.73i)7-s + (0.188 − 0.578i)8-s + (1.05 + 1.17i)9-s + (4.44 − 1.97i)10-s + (−4.18 − 0.889i)11-s + (−0.512 + 4.88i)12-s + (−0.218 − 2.07i)13-s + (7.44 − 1.58i)14-s + (−4.06 + 2.95i)15-s + (2.69 − 1.95i)16-s + (2.08 − 0.442i)17-s + ⋯
L(s)  = 1  + (−1.18 − 0.861i)2-s + (1.12 + 0.502i)3-s + (0.354 + 1.09i)4-s + (−0.525 + 0.909i)5-s + (−0.904 − 1.56i)6-s + (−0.929 + 1.03i)7-s + (0.0664 − 0.204i)8-s + (0.351 + 0.390i)9-s + (1.40 − 0.625i)10-s + (−1.26 − 0.268i)11-s + (−0.148 + 1.40i)12-s + (−0.0605 − 0.575i)13-s + (1.99 − 0.423i)14-s + (−1.04 + 0.762i)15-s + (0.672 − 0.488i)16-s + (0.504 − 0.107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $-0.984 + 0.175i$
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: \((2,\ 961,\ (\ :1/2),\ -0.984 + 0.175i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000799630 - 0.00901860i\)
\(L(\frac12)\) \(\approx\) \(0.000799630 - 0.00901860i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 + 1.21i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-1.95 - 0.870i)T + (2.00 + 2.22i)T^{2} \)
5 \( 1 + (1.17 - 2.03i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.45 - 2.73i)T + (-0.731 - 6.96i)T^{2} \)
11 \( 1 + (4.18 + 0.889i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (0.218 + 2.07i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (-2.08 + 0.442i)T + (15.5 - 6.91i)T^{2} \)
19 \( 1 + (-0.0648 + 0.616i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (-1.03 + 3.18i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (1.11 + 0.809i)T + (8.96 + 27.5i)T^{2} \)
37 \( 1 + (-0.137 - 0.237i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.90 - 1.73i)T + (27.4 - 30.4i)T^{2} \)
43 \( 1 + (-0.0281 + 0.268i)T + (-42.0 - 8.94i)T^{2} \)
47 \( 1 + (4.35 - 3.16i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (6.36 + 7.06i)T + (-5.54 + 52.7i)T^{2} \)
59 \( 1 + (5.32 + 2.37i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + 2.22T + 61T^{2} \)
67 \( 1 + (-6.80 + 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.893 + 0.992i)T + (-7.42 + 70.6i)T^{2} \)
73 \( 1 + (13.8 + 2.94i)T + (66.6 + 29.6i)T^{2} \)
79 \( 1 + (-8.48 + 1.80i)T + (72.1 - 32.1i)T^{2} \)
83 \( 1 + (-4.73 + 2.10i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (1.54 + 4.76i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-2.07 - 6.38i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.585573357112672541266797003965, −9.031973225407497064676848024122, −8.120025375603606173255470124029, −7.75833775038761584780917894554, −6.38831031570023644717717181183, −5.19421579084647065643620429602, −3.33959779444921228160565623486, −3.03895389928699925826652887686, −2.33586002978878400633317908186, −0.00539520466278984601656687743, 1.40001502087530718483431584353, 3.04363984143525257197168734658, 4.07550105687259232044923330339, 5.42669390947107612325876910136, 6.74615344128862161710012505302, 7.40841325546332541259596470559, 7.926652158723029374498367394748, 8.546686535709306015951438035497, 9.342759419865171560202635784832, 9.957034150562003006667357668943

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