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Properties

Label 2-712-712.403-c0-0-0
Degree $2$
Conductor $712$
Sign $0.00217 + 0.999i$
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 + (0.613 − 1.64i)3-s + (0.415 − 0.909i)4-s + (0.373 + 1.71i)6-s + (0.142 + 0.989i)8-s + (−1.57 − 1.36i)9-s + (0.281 − 1.95i)11-s + (−1.24 − 1.24i)12-s + (−0.654 − 0.755i)16-s + (−0.909 + 1.41i)17-s + (2.06 + 0.296i)18-s + (0.0855 + 1.19i)19-s + (0.822 + 1.80i)22-s + (1.71 + 0.373i)24-s + (0.959 − 0.281i)25-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (0.613 − 1.64i)3-s + (0.415 − 0.909i)4-s + (0.373 + 1.71i)6-s + (0.142 + 0.989i)8-s + (−1.57 − 1.36i)9-s + (0.281 − 1.95i)11-s + (−1.24 − 1.24i)12-s + (−0.654 − 0.755i)16-s + (−0.909 + 1.41i)17-s + (2.06 + 0.296i)18-s + (0.0855 + 1.19i)19-s + (0.822 + 1.80i)22-s + (1.71 + 0.373i)24-s + (0.959 − 0.281i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00217 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00217 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.00217 + 0.999i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.00217 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7515851560\)
\(L(\frac12)\) \(\approx\) \(0.7515851560\)
\(L(1)\) 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 \( 1 + (-0.613 + 1.64i)T + (-0.755 - 0.654i)T^{2} \)
5 \( 1 + (-0.959 + 0.281i)T^{2} \)
7 \( 1 + (0.281 + 0.959i)T^{2} \)
11 \( 1 + (-0.281 + 1.95i)T + (-0.959 - 0.281i)T^{2} \)
13 \( 1 + (-0.755 - 0.654i)T^{2} \)
17 \( 1 + (0.909 - 1.41i)T + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.0855 - 1.19i)T + (-0.989 + 0.142i)T^{2} \)
23 \( 1 + (-0.989 + 0.142i)T^{2} \)
29 \( 1 + (-0.281 - 0.959i)T^{2} \)
31 \( 1 + (-0.989 - 0.142i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (1.32 - 0.494i)T + (0.755 - 0.654i)T^{2} \)
43 \( 1 + (-1.56 + 1.17i)T + (0.281 - 0.959i)T^{2} \)
47 \( 1 + (-0.654 - 0.755i)T^{2} \)
53 \( 1 + (-0.654 + 0.755i)T^{2} \)
59 \( 1 + (0.148 + 0.398i)T + (-0.755 + 0.654i)T^{2} \)
61 \( 1 + (0.540 - 0.841i)T^{2} \)
67 \( 1 + (-0.627 - 1.37i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (-0.959 - 0.281i)T^{2} \)
73 \( 1 + (-0.368 - 0.425i)T + (-0.142 + 0.989i)T^{2} \)
79 \( 1 + (-0.142 + 0.989i)T^{2} \)
83 \( 1 + (-0.148 - 0.682i)T + (-0.909 + 0.415i)T^{2} \)
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

−10.42590941626356653312882124959, −9.026909447146031571690579567990, −8.387989630863804483044316002461, −8.125941891652855746568624004173, −6.93865000781591366778755451926, −6.30858795323586015705100497639, −5.66323055052109113858823318291, −3.52858218459818191099653005821, −2.20654960844176425300557546976, −1.06365130455925400985305775541, 2.27520708070380661986929873867, 3.14113533431746785323672389963, 4.48278786803441124014085460668, 4.78798972067198893608645124043, 6.82115131374994817406844522708, 7.56823431775412822501222110583, 8.824183198511035470665139816233, 9.330251285304499287017071035740, 9.746145499182694545121292796332, 10.66886667392446840258824536712

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