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Properties

Label 2-712-712.11-c0-0-0
Degree $2$
Conductor $712$
Sign $0.530 + 0.847i$
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  + (−0.654 − 0.755i)2-s + (1.80 − 0.258i)3-s + (−0.142 + 0.989i)4-s + (−1.37 − 1.19i)6-s + (0.841 − 0.540i)8-s + (2.21 − 0.650i)9-s + (−1.41 − 0.909i)11-s + 1.81i·12-s + (−0.959 − 0.281i)16-s + (−0.857 + 0.989i)17-s + (−1.94 − 1.24i)18-s + (0.304 + 1.03i)19-s + (0.239 + 1.66i)22-s + (1.37 − 1.19i)24-s + (0.415 − 0.909i)25-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (1.80 − 0.258i)3-s + (−0.142 + 0.989i)4-s + (−1.37 − 1.19i)6-s + (0.841 − 0.540i)8-s + (2.21 − 0.650i)9-s + (−1.41 − 0.909i)11-s + 1.81i·12-s + (−0.959 − 0.281i)16-s + (−0.857 + 0.989i)17-s + (−1.94 − 1.24i)18-s + (0.304 + 1.03i)19-s + (0.239 + 1.66i)22-s + (1.37 − 1.19i)24-s + (0.415 − 0.909i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.530 + 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.145505630\)
\(L(\frac12)\) \(\approx\) \(1.145505630\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 + 0.755i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
good3 \( 1 + (-1.80 + 0.258i)T + (0.959 - 0.281i)T^{2} \)
5 \( 1 + (-0.415 + 0.909i)T^{2} \)
7 \( 1 + (0.415 - 0.909i)T^{2} \)
11 \( 1 + (1.41 + 0.909i)T + (0.415 + 0.909i)T^{2} \)
13 \( 1 + (-0.959 + 0.281i)T^{2} \)
17 \( 1 + (0.857 - 0.989i)T + (-0.142 - 0.989i)T^{2} \)
19 \( 1 + (-0.304 - 1.03i)T + (-0.841 + 0.540i)T^{2} \)
23 \( 1 + (0.841 - 0.540i)T^{2} \)
29 \( 1 + (0.415 - 0.909i)T^{2} \)
31 \( 1 + (0.841 + 0.540i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.959 + 0.281i)T^{2} \)
43 \( 1 + (0.817 - 1.27i)T + (-0.415 - 0.909i)T^{2} \)
47 \( 1 + (0.959 + 0.281i)T^{2} \)
53 \( 1 + (0.959 - 0.281i)T^{2} \)
59 \( 1 + (1.49 + 0.215i)T + (0.959 + 0.281i)T^{2} \)
61 \( 1 + (-0.654 + 0.755i)T^{2} \)
67 \( 1 + (0.273 + 1.89i)T + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (-0.415 - 0.909i)T^{2} \)
73 \( 1 + (-0.797 - 0.234i)T + (0.841 + 0.540i)T^{2} \)
79 \( 1 + (-0.841 - 0.540i)T^{2} \)
83 \( 1 + (1.49 + 1.29i)T + (0.142 + 0.989i)T^{2} \)
97 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
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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

−10.34475054926092311427533557775, −9.556128078673908412120084180233, −8.657447933968062590614102493716, −8.113206410341728676086270917144, −7.68083765351508315975698374533, −6.36862733494735740520084078971, −4.53586626886116226972092511809, −3.43157732222262826565994375778, −2.74995916188776201885232267486, −1.70795619226176437029510960185, 2.03093095829244569893584446012, 2.93206283480255289774486273203, 4.50379087406728352757426630824, 5.20731307043268728148546548009, 7.10152046275782111861945078655, 7.27457338036623099521689501536, 8.311320497082867266014172329702, 8.934527415749014975061933073286, 9.655019650991420606084284829593, 10.25663646927867494718976322990

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