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Properties

Label 2-712-712.331-c0-0-0
Degree $2$
Conductor $712$
Sign $0.622 + 0.782i$
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  + (−0.142 − 0.989i)2-s + (1.25 + 0.368i)3-s + (−0.959 + 0.281i)4-s + (0.186 − 1.29i)6-s + (0.415 + 0.909i)8-s + (0.601 + 0.386i)9-s + (0.345 − 0.755i)11-s − 1.30·12-s + (0.841 − 0.540i)16-s + (0.0405 − 0.281i)17-s + (0.297 − 0.650i)18-s + (0.698 + 0.449i)19-s + (−0.797 − 0.234i)22-s + (0.186 + 1.29i)24-s + (−0.654 + 0.755i)25-s + ⋯
L(s)  = 1  + (−0.142 − 0.989i)2-s + (1.25 + 0.368i)3-s + (−0.959 + 0.281i)4-s + (0.186 − 1.29i)6-s + (0.415 + 0.909i)8-s + (0.601 + 0.386i)9-s + (0.345 − 0.755i)11-s − 1.30·12-s + (0.841 − 0.540i)16-s + (0.0405 − 0.281i)17-s + (0.297 − 0.650i)18-s + (0.698 + 0.449i)19-s + (−0.797 − 0.234i)22-s + (0.186 + 1.29i)24-s + (−0.654 + 0.755i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.622 + 0.782i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.622 + 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.190010646\)
\(L(\frac12)\) \(\approx\) \(1.190010646\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.142 + 0.989i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
good3 \( 1 + (-1.25 - 0.368i)T + (0.841 + 0.540i)T^{2} \)
5 \( 1 + (0.654 - 0.755i)T^{2} \)
7 \( 1 + (0.654 - 0.755i)T^{2} \)
11 \( 1 + (-0.345 + 0.755i)T + (-0.654 - 0.755i)T^{2} \)
13 \( 1 + (-0.841 - 0.540i)T^{2} \)
17 \( 1 + (-0.0405 + 0.281i)T + (-0.959 - 0.281i)T^{2} \)
19 \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \)
23 \( 1 + (-0.415 - 0.909i)T^{2} \)
29 \( 1 + (0.654 - 0.755i)T^{2} \)
31 \( 1 + (-0.415 + 0.909i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.91 - 0.563i)T + (0.841 - 0.540i)T^{2} \)
43 \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (-0.841 + 0.540i)T^{2} \)
53 \( 1 + (-0.841 - 0.540i)T^{2} \)
59 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
61 \( 1 + (0.142 - 0.989i)T^{2} \)
67 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
71 \( 1 + (0.654 + 0.755i)T^{2} \)
73 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
79 \( 1 + (-0.415 + 0.909i)T^{2} \)
83 \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \)
97 \( 1 + (-0.698 - 1.53i)T + (-0.654 + 0.755i)T^{2} \)
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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

−10.29829982623371478640165043711, −9.629697591498970878286645030122, −8.947200210867108794053825681606, −8.276232896255895890597842077080, −7.46824394133979216320472151609, −5.86530410198500265734434016369, −4.64861233789756483920775232919, −3.52056049804484945678933927214, −3.05212854071229786676559595183, −1.67365923784166771188416425982, 1.80711825186729283861718714979, 3.29016378712050823822261355823, 4.35204473562697950408046640380, 5.47372403399733601554554735782, 6.69307085510735417624787415857, 7.34967866036938906629110355826, 8.162834018412822838870528886126, 8.799998435442587017513476209521, 9.598083749700947419735927157955, 10.28897978433208613963442476255

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