Properties

Label 2-712-712.435-c0-0-0
Degree $2$
Conductor $712$
Sign $0.140 - 0.990i$
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  + (−0.415 + 0.909i)2-s + (0.0303 + 0.424i)3-s + (−0.654 − 0.755i)4-s + (−0.398 − 0.148i)6-s + (0.959 − 0.281i)8-s + (0.810 − 0.116i)9-s + (0.540 + 0.158i)11-s + (0.300 − 0.300i)12-s + (−0.142 + 0.989i)16-s + (0.755 − 0.345i)17-s + (−0.230 + 0.786i)18-s + (−0.574 + 0.767i)19-s + (−0.368 + 0.425i)22-s + (0.148 + 0.398i)24-s + (−0.841 + 0.540i)25-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (0.0303 + 0.424i)3-s + (−0.654 − 0.755i)4-s + (−0.398 − 0.148i)6-s + (0.959 − 0.281i)8-s + (0.810 − 0.116i)9-s + (0.540 + 0.158i)11-s + (0.300 − 0.300i)12-s + (−0.142 + 0.989i)16-s + (0.755 − 0.345i)17-s + (−0.230 + 0.786i)18-s + (−0.574 + 0.767i)19-s + (−0.368 + 0.425i)22-s + (0.148 + 0.398i)24-s + (−0.841 + 0.540i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.140 - 0.990i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (435, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.140 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8042515051\)
\(L(\frac12)\) \(\approx\) \(0.8042515051\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 - 0.909i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
good3 \( 1 + (-0.0303 - 0.424i)T + (-0.989 + 0.142i)T^{2} \)
5 \( 1 + (0.841 - 0.540i)T^{2} \)
7 \( 1 + (0.540 + 0.841i)T^{2} \)
11 \( 1 + (-0.540 - 0.158i)T + (0.841 + 0.540i)T^{2} \)
13 \( 1 + (-0.989 + 0.142i)T^{2} \)
17 \( 1 + (-0.755 + 0.345i)T + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (0.574 - 0.767i)T + (-0.281 - 0.959i)T^{2} \)
23 \( 1 + (-0.281 - 0.959i)T^{2} \)
29 \( 1 + (-0.540 - 0.841i)T^{2} \)
31 \( 1 + (-0.281 + 0.959i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-1.41 - 0.100i)T + (0.989 + 0.142i)T^{2} \)
43 \( 1 + (0.613 - 0.334i)T + (0.540 - 0.841i)T^{2} \)
47 \( 1 + (-0.142 + 0.989i)T^{2} \)
53 \( 1 + (-0.142 - 0.989i)T^{2} \)
59 \( 1 + (-0.133 + 1.86i)T + (-0.989 - 0.142i)T^{2} \)
61 \( 1 + (-0.909 + 0.415i)T^{2} \)
67 \( 1 + (1.29 - 1.49i)T + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (-0.153 + 1.07i)T + (-0.959 - 0.281i)T^{2} \)
79 \( 1 + (-0.959 - 0.281i)T^{2} \)
83 \( 1 + (0.133 + 0.0498i)T + (0.755 + 0.654i)T^{2} \)
97 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
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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

−10.44305517050797244561031152613, −9.792064896156083646552429227364, −9.219419113858530429279147584910, −8.149017229659542833125751371391, −7.39169983728840769609197728709, −6.50539691885751740398576186369, −5.57644433397889883064044278299, −4.53707171021504311585329507314, −3.67471620500386506804353531061, −1.56677321841894246317455430573, 1.28365033677039488142098374428, 2.47807395129496890565234716984, 3.80511919504140018462646590058, 4.63309875403591254988319334254, 6.07326583560129102862799010858, 7.20172425587670864115311572190, 7.934176812232318929609217461661, 8.853478023543260514931437361722, 9.696069010348392391412579928828, 10.42138115001648699218927077070

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