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Properties

Label 2-712-712.627-c0-0-0
Degree $2$
Conductor $712$
Sign $-0.917 + 0.397i$
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  + (0.841 − 0.540i)2-s + (−0.797 − 1.74i)3-s + (0.415 − 0.909i)4-s + (−1.61 − 1.03i)6-s + (−0.142 − 0.989i)8-s + (−1.75 + 2.02i)9-s + (0.0405 − 0.281i)11-s − 1.91·12-s + (−0.654 − 0.755i)16-s + (1.41 + 0.909i)17-s + (−0.381 + 2.65i)18-s + (0.186 − 0.215i)19-s + (−0.118 − 0.258i)22-s + (−1.61 + 1.03i)24-s + (−0.959 + 0.281i)25-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (−0.797 − 1.74i)3-s + (0.415 − 0.909i)4-s + (−1.61 − 1.03i)6-s + (−0.142 − 0.989i)8-s + (−1.75 + 2.02i)9-s + (0.0405 − 0.281i)11-s − 1.91·12-s + (−0.654 − 0.755i)16-s + (1.41 + 0.909i)17-s + (−0.381 + 2.65i)18-s + (0.186 − 0.215i)19-s + (−0.118 − 0.258i)22-s + (−1.61 + 1.03i)24-s + (−0.959 + 0.281i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.917 + 0.397i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (627, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ -0.917 + 0.397i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.134412823\)
\(L(\frac12)\) \(\approx\) \(1.134412823\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.841 + 0.540i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
good3 \( 1 + (0.797 + 1.74i)T + (-0.654 + 0.755i)T^{2} \)
5 \( 1 + (0.959 - 0.281i)T^{2} \)
7 \( 1 + (0.959 - 0.281i)T^{2} \)
11 \( 1 + (-0.0405 + 0.281i)T + (-0.959 - 0.281i)T^{2} \)
13 \( 1 + (0.654 - 0.755i)T^{2} \)
17 \( 1 + (-1.41 - 0.909i)T + (0.415 + 0.909i)T^{2} \)
19 \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \)
23 \( 1 + (0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.959 - 0.281i)T^{2} \)
31 \( 1 + (0.142 - 0.989i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.830 + 1.81i)T + (-0.654 - 0.755i)T^{2} \)
43 \( 1 + (0.239 - 1.66i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (0.654 + 0.755i)T^{2} \)
53 \( 1 + (0.654 - 0.755i)T^{2} \)
59 \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \)
61 \( 1 + (-0.841 - 0.540i)T^{2} \)
67 \( 1 + (0.544 + 1.19i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (0.959 + 0.281i)T^{2} \)
73 \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.142 - 0.989i)T^{2} \)
83 \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \)
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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

−10.75891729443563247670520024473, −9.652997038225391323932038304912, −8.165527178296760552476555663064, −7.45611319902412669085363121440, −6.45795400624503243708665348284, −5.84679254938929211643982009883, −5.11002548179873020738446774658, −3.51714542184789561285887230215, −2.21048062695438297806811925026, −1.15159487801728084134311645291, 3.00455720293586337967634438441, 3.87865399898078605144766319093, 4.73673287776165265692662913363, 5.48488118024485517593954771507, 6.13566010661843687958091053115, 7.35345734396281193051523291099, 8.461111653165099738945768610759, 9.553524260358764355812913582710, 10.10558416018334811208177448591, 11.12940586020430749520217750869

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