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Properties

Label 2-712-712.395-c0-0-0
Degree $2$
Conductor $712$
Sign $0.984 + 0.172i$
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  + (0.415 + 0.909i)2-s + (−1.10 − 1.27i)3-s + (−0.654 + 0.755i)4-s + (0.698 − 1.53i)6-s + (−0.959 − 0.281i)8-s + (−0.260 + 1.81i)9-s + (1.84 − 0.540i)11-s + 1.68·12-s + (−0.142 − 0.989i)16-s + (0.345 − 0.755i)17-s + (−1.75 + 0.515i)18-s + (0.273 − 1.89i)19-s + (1.25 + 1.45i)22-s + (0.698 + 1.53i)24-s + (0.841 + 0.540i)25-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)2-s + (−1.10 − 1.27i)3-s + (−0.654 + 0.755i)4-s + (0.698 − 1.53i)6-s + (−0.959 − 0.281i)8-s + (−0.260 + 1.81i)9-s + (1.84 − 0.540i)11-s + 1.68·12-s + (−0.142 − 0.989i)16-s + (0.345 − 0.755i)17-s + (−1.75 + 0.515i)18-s + (0.273 − 1.89i)19-s + (1.25 + 1.45i)22-s + (0.698 + 1.53i)24-s + (0.841 + 0.540i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.984 + 0.172i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.984 + 0.172i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8087213915\)
\(L(\frac12)\) \(\approx\) \(0.8087213915\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 - 0.909i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
good3 \( 1 + (1.10 + 1.27i)T + (-0.142 + 0.989i)T^{2} \)
5 \( 1 + (-0.841 - 0.540i)T^{2} \)
7 \( 1 + (-0.841 - 0.540i)T^{2} \)
11 \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.142 - 0.989i)T^{2} \)
17 \( 1 + (-0.345 + 0.755i)T + (-0.654 - 0.755i)T^{2} \)
19 \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \)
23 \( 1 + (0.959 + 0.281i)T^{2} \)
29 \( 1 + (-0.841 - 0.540i)T^{2} \)
31 \( 1 + (0.959 - 0.281i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.30 - 1.51i)T + (-0.142 - 0.989i)T^{2} \)
43 \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 + (0.142 + 0.989i)T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \)
61 \( 1 + (-0.415 + 0.909i)T^{2} \)
67 \( 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (-0.841 + 0.540i)T^{2} \)
73 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
79 \( 1 + (0.959 - 0.281i)T^{2} \)
83 \( 1 + (0.544 - 1.19i)T + (-0.654 - 0.755i)T^{2} \)
97 \( 1 + (-0.273 - 0.0801i)T + (0.841 + 0.540i)T^{2} \)
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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

−11.07631637880200148114978022898, −9.430307114991187556488583493888, −8.727046719439326502479411222935, −7.57798652109391957475075850114, −6.75669480780390164991816341901, −6.51092618719336287092938970249, −5.43002126751151767725305623390, −4.60970154857928719592728625317, −3.09418354843629659759596006517, −1.06949721861341796834878220543, 1.53299712250101336412350124626, 3.66459011618508755048608459373, 3.99004821180041182528631374807, 5.06037876520149224652561533268, 5.89315110511703219679009049763, 6.66917924313449610972633469244, 8.536395538938519556273811742952, 9.385892421712434965113084258628, 10.16763733186725701567979825497, 10.49685784458442842337488737779

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