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Properties

Label 2-588-21.17-c3-0-30
Degree $2$
Conductor $588$
Sign $-0.716 + 0.697i$
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  + (−1.93 − 4.82i)3-s + (3.83 + 6.63i)5-s + (−19.4 + 18.6i)9-s + (−1.13 − 0.657i)11-s + 23.9i·13-s + (24.5 − 31.3i)15-s + (29.5 − 51.1i)17-s + (−24.4 + 14.1i)19-s + (−65.7 + 37.9i)23-s + (33.1 − 57.3i)25-s + (127. + 57.6i)27-s − 302. i·29-s + (−80.9 − 46.7i)31-s + (−0.960 + 6.76i)33-s + (−133. − 231. i)37-s + ⋯
L(s)  = 1  + (−0.373 − 0.927i)3-s + (0.342 + 0.593i)5-s + (−0.721 + 0.692i)9-s + (−0.0312 − 0.0180i)11-s + 0.511i·13-s + (0.422 − 0.539i)15-s + (0.421 − 0.730i)17-s + (−0.295 + 0.170i)19-s + (−0.595 + 0.344i)23-s + (0.264 − 0.458i)25-s + (0.911 + 0.410i)27-s − 1.93i·29-s + (−0.469 − 0.270i)31-s + (−0.00506 + 0.0356i)33-s + (−0.592 − 1.02i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $-0.716 + 0.697i$
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: \((2,\ 588,\ (\ :3/2),\ -0.716 + 0.697i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9942253237\)
\(L(\frac12)\) \(\approx\) \(0.9942253237\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.93 + 4.82i)T \)
7 \( 1 \)
good5 \( 1 + (-3.83 - 6.63i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (1.13 + 0.657i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 23.9iT - 2.19e3T^{2} \)
17 \( 1 + (-29.5 + 51.1i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (24.4 - 14.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (65.7 - 37.9i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 302. iT - 2.43e4T^{2} \)
31 \( 1 + (80.9 + 46.7i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (133. + 231. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 142.T + 6.89e4T^{2} \)
43 \( 1 - 284.T + 7.95e4T^{2} \)
47 \( 1 + (-104. - 181. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (545. + 314. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (365. - 632. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-471. + 272. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-240. + 416. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 46.5iT - 3.57e5T^{2} \)
73 \( 1 + (834. + 481. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (630. + 1.09e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 841.T + 5.71e5T^{2} \)
89 \( 1 + (641. + 1.11e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 60.2iT - 9.12e5T^{2} \)
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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

−10.04482151052292472488931221204, −9.093225507380778113135898412367, −7.915953784919336091781978494304, −7.27401321978065520951884777165, −6.28391309877355702659285460812, −5.68748087370114297303748772121, −4.31983953306208645999932964764, −2.79668725983549707390875187100, −1.84939272370696029819046968394, −0.31702287366419169593337501914, 1.28976444465721412830915101497, 3.02529783983119549414170896596, 4.11386457500128775679026703295, 5.15197484237339126709449512883, 5.76601934949868446676857850990, 6.90387006212063989190654728875, 8.278203469699596840239559216974, 8.937783906561681495987105694289, 9.818390366122142780201410943540, 10.57292109879873921705633707616

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