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Properties

Label 2-31e2-31.15-c0-0-0
Degree $2$
Conductor $961$
Sign $0.800 - 0.599i$
Analytic cond. $0.479601$
Root an. cond. $0.692532$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)2-s − 5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (0.309 + 0.951i)14-s + (0.809 + 0.587i)16-s + (0.809 + 0.587i)18-s + (0.809 − 0.587i)19-s + (0.309 − 0.951i)35-s + (0.309 − 0.951i)38-s + (−0.309 − 0.951i)40-s + (0.809 − 0.587i)41-s + (−0.309 − 0.951i)45-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s − 5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (0.309 + 0.951i)9-s + (−0.809 + 0.587i)10-s + (0.309 + 0.951i)14-s + (0.809 + 0.587i)16-s + (0.809 + 0.587i)18-s + (0.809 − 0.587i)19-s + (0.309 − 0.951i)35-s + (0.309 − 0.951i)38-s + (−0.309 − 0.951i)40-s + (0.809 − 0.587i)41-s + (−0.309 − 0.951i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(961\)    =    \(31^{2}\)
Sign: $0.800 - 0.599i$
Analytic conductor: \(0.479601\)
Root analytic conductor: \(0.692532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{961} (573, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 961,\ (\ :0),\ 0.800 - 0.599i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.218300149\)
\(L(\frac12)\) \(\approx\) \(1.218300149\)
\(L(1)\) 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 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
3 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (-0.309 + 0.951i)T^{2} \)
47 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
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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

−10.62321659388754841866965603933, −9.520881951687328372734467396747, −8.500118979968597322051741501239, −7.892758070131895069074756497659, −7.03021885067196271822906544245, −5.61640950188835646776086368368, −4.92070586494791923869689276324, −3.99992463075727683850497885626, −3.07408450536182380060690743409, −2.12279685446713081119735335241, 0.990534801976033516665361488151, 3.41600072498515458727613160522, 3.91402161414175759583179288958, 4.74563035446808683017939455733, 5.90119211391729129107191851526, 6.74687668537547120398363177976, 7.36202380002155801657416394872, 8.162678412961214099815384512034, 9.545013209075178272523631630261, 9.972325190319962556273292130807

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