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Properties

Label 2-712-712.643-c0-0-0
Degree $2$
Conductor $712$
Sign $-0.833 + 0.551i$
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 + (−0.125 − 0.0683i)3-s + (−0.959 − 0.281i)4-s + (−0.0855 + 0.114i)6-s + (−0.415 + 0.909i)8-s + (−0.529 − 0.824i)9-s + (−0.755 − 1.65i)11-s + (0.100 + 0.100i)12-s + (0.841 + 0.540i)16-s + (−0.281 + 0.0405i)17-s + (−0.891 + 0.406i)18-s + (0.398 − 1.83i)19-s + (−1.74 + 0.512i)22-s + (0.114 − 0.0855i)24-s + (0.654 + 0.755i)25-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)2-s + (−0.125 − 0.0683i)3-s + (−0.959 − 0.281i)4-s + (−0.0855 + 0.114i)6-s + (−0.415 + 0.909i)8-s + (−0.529 − 0.824i)9-s + (−0.755 − 1.65i)11-s + (0.100 + 0.100i)12-s + (0.841 + 0.540i)16-s + (−0.281 + 0.0405i)17-s + (−0.891 + 0.406i)18-s + (0.398 − 1.83i)19-s + (−1.74 + 0.512i)22-s + (0.114 − 0.0855i)24-s + (0.654 + 0.755i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.833 + 0.551i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ -0.833 + 0.551i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7423501918\)
\(L(\frac12)\) \(\approx\) \(0.7423501918\)
\(L(1)\) 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 \( 1 + (0.125 + 0.0683i)T + (0.540 + 0.841i)T^{2} \)
5 \( 1 + (-0.654 - 0.755i)T^{2} \)
7 \( 1 + (-0.755 + 0.654i)T^{2} \)
11 \( 1 + (0.755 + 1.65i)T + (-0.654 + 0.755i)T^{2} \)
13 \( 1 + (0.540 + 0.841i)T^{2} \)
17 \( 1 + (0.281 - 0.0405i)T + (0.959 - 0.281i)T^{2} \)
19 \( 1 + (-0.398 + 1.83i)T + (-0.909 - 0.415i)T^{2} \)
23 \( 1 + (-0.909 - 0.415i)T^{2} \)
29 \( 1 + (0.755 - 0.654i)T^{2} \)
31 \( 1 + (-0.909 + 0.415i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-0.677 - 1.24i)T + (-0.540 + 0.841i)T^{2} \)
43 \( 1 + (0.559 - 1.50i)T + (-0.755 - 0.654i)T^{2} \)
47 \( 1 + (0.841 + 0.540i)T^{2} \)
53 \( 1 + (0.841 - 0.540i)T^{2} \)
59 \( 1 + (1.05 - 0.574i)T + (0.540 - 0.841i)T^{2} \)
61 \( 1 + (-0.989 + 0.142i)T^{2} \)
67 \( 1 + (-1.03 + 0.304i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (-0.654 + 0.755i)T^{2} \)
73 \( 1 + (-1.27 - 0.817i)T + (0.415 + 0.909i)T^{2} \)
79 \( 1 + (0.415 + 0.909i)T^{2} \)
83 \( 1 + (-1.05 + 1.40i)T + (-0.281 - 0.959i)T^{2} \)
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

−10.56091108754862457020479557324, −9.347779910749113871059256773379, −8.879691904724648004238137853189, −7.974094478997200711107833178478, −6.54490125471138568499543948464, −5.62071029563005759945161633646, −4.75073559961586715626077360906, −3.33960154525499314591466226106, −2.76925680504148148954995862220, −0.802986117586877474710365817099, 2.26117913916977559249081262239, 3.86355917082105941296074256080, 4.91046596995474687217018805061, 5.52319082732450370479386437064, 6.63177320979800702610213684118, 7.60989826201530110485240627061, 8.074820323229112983254830674128, 9.152495431730827127108297479221, 10.10544485972343241763131388896, 10.66495244103570130021130627964

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