Properties

Label 2-588-21.17-c3-0-27
Degree $2$
Conductor $588$
Sign $0.997 + 0.0633i$
Analytic cond. $34.6931$
Root an. cond. $5.89008$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.5 + 2.59i)3-s + (13.5 + 23.3i)9-s − 91.7i·13-s + (109.5 − 63.2i)19-s + (62.5 − 108. i)25-s + 140. i·27-s + (163.5 + 94.3i)31-s + (161.5 + 279. i)37-s + (238.5 − 413. i)39-s − 71·43-s + 657·57-s + (810 − 467. i)61-s + (63.5 − 109. i)67-s + (−1.05e3 − 608. i)73-s + (562.5 − 324. i)75-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (0.5 + 0.866i)9-s − 1.95i·13-s + (1.32 − 0.763i)19-s + (0.5 − 0.866i)25-s + 1.00i·27-s + (0.947 + 0.546i)31-s + (0.717 + 1.24i)37-s + (0.979 − 1.69i)39-s − 0.251·43-s + 1.52·57-s + (1.70 − 0.981i)61-s + (0.115 − 0.200i)67-s + (−1.69 − 0.976i)73-s + (0.866 − 0.499i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.997 + 0.0633i$
Analytic conductor: \(34.6931\)
Root analytic conductor: \(5.89008\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{588} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 588,\ (\ :3/2),\ 0.997 + 0.0633i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.881265942\)
\(L(\frac12)\) \(\approx\) \(2.881265942\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.5 - 2.59i)T \)
7 \( 1 \)
good5 \( 1 + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 91.7iT - 2.19e3T^{2} \)
17 \( 1 + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-109.5 + 63.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 2.43e4T^{2} \)
31 \( 1 + (-163.5 - 94.3i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-161.5 - 279. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 + 71T + 7.95e4T^{2} \)
47 \( 1 + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-810 + 467. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-63.5 + 109. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 3.57e5T^{2} \)
73 \( 1 + (1.05e3 + 608. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-693.5 - 1.20e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.37e3iT - 9.12e5T^{2} \)
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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

−10.15189739663851617241347624453, −9.520159408655716459250819644105, −8.370687232438488450276617280434, −7.932290843983537361536899296417, −6.83205218331298574174075091957, −5.45582441562956184295837161965, −4.66991210030904743473878505478, −3.30159839267311491291017389863, −2.68574186266894372197806186372, −0.908492470768639451528875965591, 1.19175388556090003212858804359, 2.26440688549531601176972248585, 3.51144314592946676026291983126, 4.45526585299765705162072649143, 5.88888591498044158134754085528, 6.94152364794610827115624567011, 7.54282890221922682289234963078, 8.626722009744932672357138763792, 9.326174496023494079955359574699, 9.998561139831442594067362524674

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