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Properties

Label 2-588-21.17-c3-0-22
Degree $2$
Conductor $588$
Sign $-0.130 - 0.991i$
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  + (4.58 + 2.43i)3-s + (9.23 + 16.0i)5-s + (15.1 + 22.3i)9-s + (36.0 + 20.7i)11-s + 43.5i·13-s + (3.38 + 95.9i)15-s + (62.7 − 108. i)17-s + (59.2 − 34.2i)19-s + (−77.2 + 44.5i)23-s + (−108. + 187. i)25-s + (14.8 + 139. i)27-s − 133. i·29-s + (−75.4 − 43.5i)31-s + (114. + 183. i)33-s + (114. + 197. i)37-s + ⋯
L(s)  = 1  + (0.883 + 0.469i)3-s + (0.826 + 1.43i)5-s + (0.559 + 0.828i)9-s + (0.987 + 0.569i)11-s + 0.928i·13-s + (0.0583 + 1.65i)15-s + (0.895 − 1.55i)17-s + (0.715 − 0.412i)19-s + (−0.700 + 0.404i)23-s + (−0.865 + 1.49i)25-s + (0.105 + 0.994i)27-s − 0.857i·29-s + (−0.437 − 0.252i)31-s + (0.604 + 0.966i)33-s + (0.507 + 0.878i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.130 - 0.991i$
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: \((2,\ 588,\ (\ :3/2),\ -0.130 - 0.991i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.545449691\)
\(L(\frac12)\) \(\approx\) \(3.545449691\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.58 - 2.43i)T \)
7 \( 1 \)
good5 \( 1 + (-9.23 - 16.0i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-36.0 - 20.7i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 43.5iT - 2.19e3T^{2} \)
17 \( 1 + (-62.7 + 108. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-59.2 + 34.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (77.2 - 44.5i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 133. iT - 2.43e4T^{2} \)
31 \( 1 + (75.4 + 43.5i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-114. - 197. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 37.0T + 6.89e4T^{2} \)
43 \( 1 - 412.T + 7.95e4T^{2} \)
47 \( 1 + (303. + 524. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (417. + 241. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (43.6 - 75.5i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (655. - 378. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-17.3 + 30.0i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 481. iT - 3.57e5T^{2} \)
73 \( 1 + (496. + 286. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (227. + 393. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 866.T + 5.71e5T^{2} \)
89 \( 1 + (-309. - 535. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 593. iT - 9.12e5T^{2} \)
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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

−10.20869455138770618780785269189, −9.594153765451193426659091424384, −9.218289965102372559105540346270, −7.65623857433533030300985713112, −7.08171088890311566510756548249, −6.13881374214623865835498306463, −4.82194550652085007337422212880, −3.63092482016341432050377989819, −2.72811837263136176299246339198, −1.75006318880054227740769791854, 1.02326026758385659602282087599, 1.64621152233961179523136507941, 3.22670957900105765463898696819, 4.25603426548519136610072940390, 5.65087854533660577816919107798, 6.20165193599321114140949394937, 7.73930791575688979021917479127, 8.322519623605891508327114635914, 9.147484115417281132730636523359, 9.698499711572113340430782007274

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