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Properties

Label 2-588-21.17-c3-0-19
Degree $2$
Conductor $588$
Sign $0.979 + 0.202i$
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  + (−4.40 − 2.75i)3-s + (9.23 + 16.0i)5-s + (11.8 + 24.2i)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 + (101. + 190. i)33-s + (114. + 197. i)37-s + ⋯
L(s)  = 1  + (−0.847 − 0.530i)3-s + (0.826 + 1.43i)5-s + (0.437 + 0.899i)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 + (0.534 + 1.00i)33-s + (0.507 + 0.878i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(588\)    =    \(2^{2} \cdot 3 \cdot 7^{2}\)
Sign: $0.979 + 0.202i$
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: \((2,\ 588,\ (\ :3/2),\ 0.979 + 0.202i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.559220475\)
\(L(\frac12)\) \(\approx\) \(1.559220475\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.40 + 2.75i)T \)
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

−10.46336627018360100404485898960, −9.800646529598372848643830615929, −8.259739892339714020781118675408, −7.33071443911811740507918067852, −6.66066542377696042544843129641, −5.70549283722128713510133029704, −5.11856820922845275310185222419, −3.14212862996421332609859493553, −2.39245417897717304064676283326, −0.71724132473300231236826092649, 0.859451007495310784461762445518, 2.06650753402294774761510549025, 4.06243041384964214476888322472, 4.80465837276549637975190117456, 5.63248919538773665032668453054, 6.33817973610329543036728825835, 7.73885773924656264274750473295, 8.806872014485077921931496648869, 9.548487281016339869251264545322, 10.20552591771547519831832853536

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