Properties

Label 2-712-712.195-c0-0-0
Degree $2$
Conductor $712$
Sign $0.961 - 0.274i$
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  + (0.959 + 0.281i)2-s + (−1.56 − 0.340i)3-s + (0.841 + 0.540i)4-s + (−1.40 − 0.767i)6-s + (0.654 + 0.755i)8-s + (1.42 + 0.649i)9-s + (0.989 − 1.14i)11-s + (−1.13 − 1.13i)12-s + (0.415 + 0.909i)16-s + (0.540 + 1.84i)17-s + (1.18 + 1.02i)18-s + (−0.0498 − 0.133i)19-s + (1.27 − 0.817i)22-s + (−0.767 − 1.40i)24-s + (0.142 − 0.989i)25-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)2-s + (−1.56 − 0.340i)3-s + (0.841 + 0.540i)4-s + (−1.40 − 0.767i)6-s + (0.654 + 0.755i)8-s + (1.42 + 0.649i)9-s + (0.989 − 1.14i)11-s + (−1.13 − 1.13i)12-s + (0.415 + 0.909i)16-s + (0.540 + 1.84i)17-s + (1.18 + 1.02i)18-s + (−0.0498 − 0.133i)19-s + (1.27 − 0.817i)22-s + (−0.767 − 1.40i)24-s + (0.142 − 0.989i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.961 - 0.274i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.961 - 0.274i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.119652466\)
\(L(\frac12)\) \(\approx\) \(1.119652466\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 - 0.281i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
good3 \( 1 + (1.56 + 0.340i)T + (0.909 + 0.415i)T^{2} \)
5 \( 1 + (-0.142 + 0.989i)T^{2} \)
7 \( 1 + (0.989 + 0.142i)T^{2} \)
11 \( 1 + (-0.989 + 1.14i)T + (-0.142 - 0.989i)T^{2} \)
13 \( 1 + (0.909 + 0.415i)T^{2} \)
17 \( 1 + (-0.540 - 1.84i)T + (-0.841 + 0.540i)T^{2} \)
19 \( 1 + (0.0498 + 0.133i)T + (-0.755 + 0.654i)T^{2} \)
23 \( 1 + (-0.755 + 0.654i)T^{2} \)
29 \( 1 + (-0.989 - 0.142i)T^{2} \)
31 \( 1 + (-0.755 - 0.654i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (0.300 + 1.38i)T + (-0.909 + 0.415i)T^{2} \)
43 \( 1 + (0.956 - 0.0683i)T + (0.989 - 0.142i)T^{2} \)
47 \( 1 + (0.415 + 0.909i)T^{2} \)
53 \( 1 + (0.415 - 0.909i)T^{2} \)
59 \( 1 + (1.71 - 0.373i)T + (0.909 - 0.415i)T^{2} \)
61 \( 1 + (0.281 + 0.959i)T^{2} \)
67 \( 1 + (1.53 - 0.983i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (0.822 + 1.80i)T + (-0.654 + 0.755i)T^{2} \)
79 \( 1 + (-0.654 + 0.755i)T^{2} \)
83 \( 1 + (-1.71 - 0.936i)T + (0.540 + 0.841i)T^{2} \)
97 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
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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

−10.83162822655626555551467679038, −10.42121431977304396879603600886, −8.739427705581632783067141273778, −7.79264507893330704996754459971, −6.58060827455542930515800145641, −6.20734696635415459553122507369, −5.51419199970349441722612833531, −4.43015340808944021429026630559, −3.45093070018112408333418040894, −1.53958419053261519134445723691, 1.45395325214318122294701810424, 3.24446801017868137171227324475, 4.59585556547170963824532285668, 4.93890715313061149833338254339, 5.98031035601500526475826149918, 6.77818294169547876524788746683, 7.44715000672715621521866858021, 9.460849002793106407677399265827, 9.908986562507884776223214437906, 10.91655015923820350064467187056

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