Properties

Label 2-712-712.603-c0-0-0
Degree $2$
Conductor $712$
Sign $-0.948 + 0.316i$
Analytic cond. $0.355334$
Root an. cond. $0.596099$
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  + (0.142 − 0.989i)2-s + (0.956 − 1.75i)3-s + (−0.959 − 0.281i)4-s + (−1.59 − 1.19i)6-s + (−0.415 + 0.909i)8-s + (−1.61 − 2.50i)9-s + (0.755 + 1.65i)11-s + (−1.41 + 1.41i)12-s + (0.841 + 0.540i)16-s + (0.281 − 0.0405i)17-s + (−2.71 + 1.23i)18-s + (−0.682 − 0.148i)19-s + (1.74 − 0.512i)22-s + (1.19 + 1.59i)24-s + (0.654 + 0.755i)25-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)2-s + (0.956 − 1.75i)3-s + (−0.959 − 0.281i)4-s + (−1.59 − 1.19i)6-s + (−0.415 + 0.909i)8-s + (−1.61 − 2.50i)9-s + (0.755 + 1.65i)11-s + (−1.41 + 1.41i)12-s + (0.841 + 0.540i)16-s + (0.281 − 0.0405i)17-s + (−2.71 + 1.23i)18-s + (−0.682 − 0.148i)19-s + (1.74 − 0.512i)22-s + (1.19 + 1.59i)24-s + (0.654 + 0.755i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.948 + 0.316i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (603, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ -0.948 + 0.316i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.209398615\)
\(L(\frac12)\) \(\approx\) \(1.209398615\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 + 0.989i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
good3 \( 1 + (-0.956 + 1.75i)T + (-0.540 - 0.841i)T^{2} \)
5 \( 1 + (-0.654 - 0.755i)T^{2} \)
7 \( 1 + (0.755 - 0.654i)T^{2} \)
11 \( 1 + (-0.755 - 1.65i)T + (-0.654 + 0.755i)T^{2} \)
13 \( 1 + (-0.540 - 0.841i)T^{2} \)
17 \( 1 + (-0.281 + 0.0405i)T + (0.959 - 0.281i)T^{2} \)
19 \( 1 + (0.682 + 0.148i)T + (0.909 + 0.415i)T^{2} \)
23 \( 1 + (0.909 + 0.415i)T^{2} \)
29 \( 1 + (-0.755 + 0.654i)T^{2} \)
31 \( 1 + (0.909 - 0.415i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-1.24 + 0.677i)T + (0.540 - 0.841i)T^{2} \)
43 \( 1 + (1.12 + 0.418i)T + (0.755 + 0.654i)T^{2} \)
47 \( 1 + (0.841 + 0.540i)T^{2} \)
53 \( 1 + (0.841 - 0.540i)T^{2} \)
59 \( 1 + (-0.767 - 1.40i)T + (-0.540 + 0.841i)T^{2} \)
61 \( 1 + (0.989 - 0.142i)T^{2} \)
67 \( 1 + (1.03 - 0.304i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (1.27 + 0.817i)T + (0.415 + 0.909i)T^{2} \)
79 \( 1 + (0.415 + 0.909i)T^{2} \)
83 \( 1 + (0.767 + 0.574i)T + (0.281 + 0.959i)T^{2} \)
97 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
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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

−10.16986485399117671714461882467, −9.190555340683690116329566791294, −8.740725111775512945294323799375, −7.61496319354242698656534041652, −6.97480694454730868232656185967, −5.92376518956500961461503153919, −4.41225386745073748852186976847, −3.24675227796918245242068580769, −2.19741629301470957274176104910, −1.39761636828549573603735685266, 2.99359073570468077692913126285, 3.76100272101966714409052296298, 4.57594784130926744471328020487, 5.53628234508719717029553377609, 6.43714969983373998210825363886, 8.014552460806349587720818443424, 8.469194159277611648575734733408, 9.118360405054689014649652018667, 9.884435353057475952080128920114, 10.72813345331865042721830697258

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