Properties

Label 2-712-712.667-c0-0-0
Degree $2$
Conductor $712$
Sign $0.0928 - 0.995i$
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 + (−1.07 − 1.66i)3-s + (0.841 + 0.540i)4-s + (0.557 + 1.89i)6-s + (−0.654 − 0.755i)8-s + (−1.21 + 2.65i)9-s + (−0.857 + 0.989i)11-s − 1.97i·12-s + (0.415 + 0.909i)16-s + (−1.84 + 0.540i)17-s + (1.91 − 2.20i)18-s + (−1.37 − 0.627i)19-s + (1.10 − 0.708i)22-s + (−0.557 + 1.89i)24-s + (−0.142 + 0.989i)25-s + ⋯
L(s)  = 1  + (−0.959 − 0.281i)2-s + (−1.07 − 1.66i)3-s + (0.841 + 0.540i)4-s + (0.557 + 1.89i)6-s + (−0.654 − 0.755i)8-s + (−1.21 + 2.65i)9-s + (−0.857 + 0.989i)11-s − 1.97i·12-s + (0.415 + 0.909i)16-s + (−1.84 + 0.540i)17-s + (1.91 − 2.20i)18-s + (−1.37 − 0.627i)19-s + (1.10 − 0.708i)22-s + (−0.557 + 1.89i)24-s + (−0.142 + 0.989i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0928 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0928 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.0928 - 0.995i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.0928 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05726893014\)
\(L(\frac12)\) \(\approx\) \(0.05726893014\)
\(L(1)\) 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 \( 1 + (1.07 + 1.66i)T + (-0.415 + 0.909i)T^{2} \)
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 \( 1 + (1.37 + 0.627i)T + (0.654 + 0.755i)T^{2} \)
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 \( 1 + (-0.415 - 0.909i)T^{2} \)
43 \( 1 + (0.425 + 0.368i)T + (0.142 + 0.989i)T^{2} \)
47 \( 1 + (-0.415 - 0.909i)T^{2} \)
53 \( 1 + (-0.415 + 0.909i)T^{2} \)
59 \( 1 + (0.304 - 0.474i)T + (-0.415 - 0.909i)T^{2} \)
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 \( 1 + (0.304 + 1.03i)T + (-0.841 + 0.540i)T^{2} \)
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

−10.90669541640051442339804043802, −10.33026485882329984077591034794, −8.873951312493918146170227296452, −8.164211893910006635064352234615, −7.19016547691413003394056010945, −6.80944904921304737190291428071, −5.89609267485123009301570663358, −4.64166250587002288280486906020, −2.42977696132648258764099142105, −1.80061388911622004169962072277, 0.087720807979008825358061568418, 2.72847182945239707993008912618, 4.16941828939288576220831039761, 5.10380817578331401381502902163, 6.07757305683537156651168853152, 6.58176149947625357772347710532, 8.226798612440297171012698555028, 8.879994929428453489030577941061, 9.661139207139599018179503207851, 10.59500937301234215168451258406

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