Properties

Label 2-712-712.579-c0-0-0
Degree $2$
Conductor $712$
Sign $0.993 - 0.111i$
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 + (−0.239 + 0.153i)3-s + (0.841 + 0.540i)4-s + (0.273 − 0.0801i)6-s + (−0.654 − 0.755i)8-s + (−0.381 + 0.835i)9-s + (0.857 − 0.989i)11-s − 0.284·12-s + (0.415 + 0.909i)16-s + (1.84 − 0.540i)17-s + (0.601 − 0.694i)18-s + (−0.544 + 1.19i)19-s + (−1.10 + 0.708i)22-s + (0.273 + 0.0801i)24-s + (−0.142 + 0.989i)25-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (−0.239 + 0.153i)3-s + (0.841 + 0.540i)4-s + (0.273 − 0.0801i)6-s + (−0.654 − 0.755i)8-s + (−0.381 + 0.835i)9-s + (0.857 − 0.989i)11-s − 0.284·12-s + (0.415 + 0.909i)16-s + (1.84 − 0.540i)17-s + (0.601 − 0.694i)18-s + (−0.544 + 1.19i)19-s + (−1.10 + 0.708i)22-s + (0.273 + 0.0801i)24-s + (−0.142 + 0.989i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.993 - 0.111i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.993 - 0.111i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6112928944\)
\(L(\frac12)\) \(\approx\) \(0.6112928944\)
\(L(1)\) 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 \( 1 + (0.239 - 0.153i)T + (0.415 - 0.909i)T^{2} \)
5 \( 1 + (0.142 - 0.989i)T^{2} \)
7 \( 1 + (0.142 - 0.989i)T^{2} \)
11 \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \)
13 \( 1 + (-0.415 + 0.909i)T^{2} \)
17 \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \)
19 \( 1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2} \)
23 \( 1 + (0.654 + 0.755i)T^{2} \)
29 \( 1 + (0.142 - 0.989i)T^{2} \)
31 \( 1 + (0.654 - 0.755i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.68 - 1.08i)T + (0.415 + 0.909i)T^{2} \)
43 \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (-0.415 - 0.909i)T^{2} \)
53 \( 1 + (-0.415 + 0.909i)T^{2} \)
59 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
61 \( 1 + (0.959 - 0.281i)T^{2} \)
67 \( 1 + (-0.698 + 0.449i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (0.142 + 0.989i)T^{2} \)
73 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
79 \( 1 + (0.654 - 0.755i)T^{2} \)
83 \( 1 + (1.61 - 0.474i)T + (0.841 - 0.540i)T^{2} \)
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

−10.68707893138445768099470639255, −9.771919772373541953084473626522, −9.047216818414899703498621572250, −8.030687421860408147252561940694, −7.55659730135463130334456770665, −6.19043364638265820190222187371, −5.54306893715118055257733638774, −3.89664009241904208880875700200, −2.89516130504653282196291442676, −1.35933979859112496892744866388, 1.17257149839545828214491774049, 2.69010799696148106615355746845, 4.15613149177777031602974917886, 5.60504861476571166872551005976, 6.37233929788050969396029730670, 7.15697077638287343643973383513, 8.025629767184901424233806503527, 9.059551634790033165270304369138, 9.586174503266090128624390060783, 10.46066404462533833305497269800

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