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Properties

Label 2-1564-1564.679-c0-0-0
Degree $2$
Conductor $1564$
Sign $-0.506 - 0.862i$
Analytic cond. $0.780537$
Root an. cond. $0.883480$
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  + (−0.841 + 0.540i)2-s + (−0.215 + 1.49i)3-s + (0.415 − 0.909i)4-s + (−0.627 − 1.37i)6-s + (−1.19 − 1.37i)7-s + (0.142 + 0.989i)8-s + (−1.23 − 0.361i)9-s + (1.66 + 1.07i)11-s + (1.27 + 0.817i)12-s + (0.544 − 0.627i)13-s + (1.74 + 0.512i)14-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)17-s + (1.23 − 0.361i)18-s + (2.31 − 1.48i)21-s − 1.97·22-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (−0.215 + 1.49i)3-s + (0.415 − 0.909i)4-s + (−0.627 − 1.37i)6-s + (−1.19 − 1.37i)7-s + (0.142 + 0.989i)8-s + (−1.23 − 0.361i)9-s + (1.66 + 1.07i)11-s + (1.27 + 0.817i)12-s + (0.544 − 0.627i)13-s + (1.74 + 0.512i)14-s + (−0.654 − 0.755i)16-s + (0.415 + 0.909i)17-s + (1.23 − 0.361i)18-s + (2.31 − 1.48i)21-s − 1.97·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1564\)    =    \(2^{2} \cdot 17 \cdot 23\)
Sign: $-0.506 - 0.862i$
Analytic conductor: \(0.780537\)
Root analytic conductor: \(0.883480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1564} (679, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1564,\ (\ :0),\ -0.506 - 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6480785990\)
\(L(\frac12)\) \(\approx\) \(0.6480785990\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 - 0.540i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (0.909 - 0.415i)T \)
good3 \( 1 + (0.215 - 1.49i)T + (-0.959 - 0.281i)T^{2} \)
5 \( 1 + (-0.841 + 0.540i)T^{2} \)
7 \( 1 + (1.19 + 1.37i)T + (-0.142 + 0.989i)T^{2} \)
11 \( 1 + (-1.66 - 1.07i)T + (0.415 + 0.909i)T^{2} \)
13 \( 1 + (-0.544 + 0.627i)T + (-0.142 - 0.989i)T^{2} \)
19 \( 1 + (0.654 + 0.755i)T^{2} \)
29 \( 1 + (0.654 - 0.755i)T^{2} \)
31 \( 1 + (-0.281 - 1.95i)T + (-0.959 + 0.281i)T^{2} \)
37 \( 1 + (-0.841 - 0.540i)T^{2} \)
41 \( 1 + (-0.841 + 0.540i)T^{2} \)
43 \( 1 + (0.959 + 0.281i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2} \)
59 \( 1 + (0.142 + 0.989i)T^{2} \)
61 \( 1 + (0.959 - 0.281i)T^{2} \)
67 \( 1 + (-0.415 + 0.909i)T^{2} \)
71 \( 1 + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (-0.708 + 0.817i)T + (-0.142 - 0.989i)T^{2} \)
83 \( 1 + (-0.841 - 0.540i)T^{2} \)
89 \( 1 + (0.0405 - 0.281i)T + (-0.959 - 0.281i)T^{2} \)
97 \( 1 + (-0.841 + 0.540i)T^{2} \)
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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

−9.960956836705006109831518267614, −9.289259996944840760571756886114, −8.582347724750626085731206178066, −7.43022141049763583534130237046, −6.63055157221807214972393184624, −6.08632490282344653059632930413, −4.85476692785787519241514603088, −4.01304793640710428903719887239, −3.36838247486822046668333345925, −1.28450008842816667470635746831, 0.815369464374695624342045676143, 2.00349132351298894888367770286, 2.90553725311221697580194985260, 3.84907297789691696418518890764, 5.83801758270600481287992766638, 6.37111013148516578087959902872, 6.83452443400791779712701849632, 7.88047609435318572894849094514, 8.756016480795766919658224412826, 9.155526113559021407760338394565

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