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Properties

Label 2-588-21.11-c2-0-22
Degree $2$
Conductor $588$
Sign $-0.999 + 0.0130i$
Analytic cond. $16.0218$
Root an. cond. $4.00272$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.68 + 1.34i)3-s + (5.28 − 3.05i)5-s + (5.39 − 7.20i)9-s + (−14.9 − 8.63i)11-s − 19.2·13-s + (−10.0 + 15.2i)15-s + (17.8 + 10.3i)17-s + (6.09 + 10.5i)19-s + (−16.8 + 9.70i)23-s + (6.12 − 10.6i)25-s + (−4.80 + 26.5i)27-s − 3.24i·29-s + (−4.24 + 7.34i)31-s + (51.6 + 3.08i)33-s + (−33.8 − 58.6i)37-s + ⋯
L(s)  = 1  + (−0.894 + 0.447i)3-s + (1.05 − 0.610i)5-s + (0.599 − 0.800i)9-s + (−1.35 − 0.784i)11-s − 1.48·13-s + (−0.672 + 1.01i)15-s + (1.04 + 0.605i)17-s + (0.320 + 0.555i)19-s + (−0.730 + 0.421i)23-s + (0.244 − 0.424i)25-s + (−0.177 + 0.984i)27-s − 0.111i·29-s + (−0.136 + 0.237i)31-s + (1.56 + 0.0934i)33-s + (−0.915 − 1.58i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0130i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.999 + 0.0130i$
Analytic conductor: \(16.0218\)
Root analytic conductor: \(4.00272\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :1),\ -0.999 + 0.0130i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.03719786878\)
\(L(\frac12)\) \(\approx\) \(0.03719786878\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (2.68 - 1.34i)T \)
7 \( 1 \)
good5 \( 1 + (-5.28 + 3.05i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (14.9 + 8.63i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 19.2T + 169T^{2} \)
17 \( 1 + (-17.8 - 10.3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-6.09 - 10.5i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (16.8 - 9.70i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 3.24iT - 841T^{2} \)
31 \( 1 + (4.24 - 7.34i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (33.8 + 58.6i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 55.6iT - 1.68e3T^{2} \)
43 \( 1 + 49.7T + 1.84e3T^{2} \)
47 \( 1 + (39.6 - 22.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (28.9 + 16.7i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (58.8 + 33.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-9.10 - 15.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (33 - 57.1i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 93.8iT - 5.04e3T^{2} \)
73 \( 1 + (-3.88 + 6.73i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (52.4 + 90.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 51.1iT - 6.88e3T^{2} \)
89 \( 1 + (-130. + 75.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 188.T + 9.40e3T^{2} \)
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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

−9.950315130434367856528195830508, −9.634305860219528153831960696425, −8.302556900794824361994240270674, −7.35532576160448302552181463968, −5.93563666020494413025871506794, −5.51414239527947843845368729318, −4.75609321772862563114506662938, −3.23211579074574026864755347495, −1.66328398503617076785402175561, −0.01458611478463767215335791301, 1.91770421473030640120844476189, 2.79865999401609015016845296487, 4.93387748824296672378285558773, 5.28676821790424235774307777155, 6.45585723362702413313725913903, 7.24243411261509725847247017032, 7.928824931265214978468346981926, 9.667973123654176127953013631982, 10.09390173445580807038139454688, 10.69627862742268229425761279164

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