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Properties

Label 2-588-7.4-c5-0-8
Degree 22
Conductor 588588
Sign 0.198−0.980i0.198 - 0.980i
Analytic cond. 94.305694.3056
Root an. cond. 9.711119.71111
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.5 + 7.79i)3-s + (−17.6 − 30.6i)5-s + (−40.5 − 70.1i)9-s + (−21.7 + 37.7i)11-s − 648.·13-s + 318.·15-s + (299. − 518. i)17-s + (−320. − 555. i)19-s + (135. + 235. i)23-s + (936. − 1.62e3i)25-s + 729·27-s − 5.28e3·29-s + (−221. + 383. i)31-s + (−196. − 339. i)33-s + (4.44e3 + 7.69e3i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.316 − 0.548i)5-s + (−0.166 − 0.288i)9-s + (−0.0542 + 0.0939i)11-s − 1.06·13-s + 0.365·15-s + (0.251 − 0.435i)17-s + (−0.203 − 0.352i)19-s + (0.0535 + 0.0927i)23-s + (0.299 − 0.519i)25-s + 0.192·27-s − 1.16·29-s + (−0.0414 + 0.0717i)31-s + (−0.0313 − 0.0542i)33-s + (0.533 + 0.924i)37-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=((0.198−0.980i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+5/2) L(s)=((0.198−0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: 0.198−0.980i0.198 - 0.980i
Analytic conductor: 94.305694.3056
Root analytic conductor: 9.711119.71111
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ588(361,⋅)\chi_{588} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 588, ( :5/2), 0.198−0.980i)(2,\ 588,\ (\ :5/2),\ 0.198 - 0.980i)

Particular Values

L(3)L(3) ≈\approx 0.92125806480.9212580648
L(12)L(\frac12) ≈\approx 0.92125806480.9212580648
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(4.5−7.79i)T 1 + (4.5 - 7.79i)T
7 1 1
good5 1+(17.6+30.6i)T+(−1.56e3+2.70e3i)T2 1 + (17.6 + 30.6i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(21.7−37.7i)T+(−8.05e4−1.39e5i)T2 1 + (21.7 - 37.7i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1+648.T+3.71e5T2 1 + 648.T + 3.71e5T^{2}
17 1+(−299.+518.i)T+(−7.09e5−1.22e6i)T2 1 + (-299. + 518. i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(320.+555.i)T+(−1.23e6+2.14e6i)T2 1 + (320. + 555. i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(−135.−235.i)T+(−3.21e6+5.57e6i)T2 1 + (-135. - 235. i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1+5.28e3T+2.05e7T2 1 + 5.28e3T + 2.05e7T^{2}
31 1+(221.−383.i)T+(−1.43e7−2.47e7i)T2 1 + (221. - 383. i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(−4.44e3−7.69e3i)T+(−3.46e7+6.00e7i)T2 1 + (-4.44e3 - 7.69e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1+7.48e3T+1.15e8T2 1 + 7.48e3T + 1.15e8T^{2}
43 1−3.60e3T+1.47e8T2 1 - 3.60e3T + 1.47e8T^{2}
47 1+(1.50e3+2.61e3i)T+(−1.14e8+1.98e8i)T2 1 + (1.50e3 + 2.61e3i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(−3.55e3+6.16e3i)T+(−2.09e8−3.62e8i)T2 1 + (-3.55e3 + 6.16e3i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(−1.64e4+2.84e4i)T+(−3.57e8−6.19e8i)T2 1 + (-1.64e4 + 2.84e4i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(9.58e3+1.65e4i)T+(−4.22e8+7.31e8i)T2 1 + (9.58e3 + 1.65e4i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(7.49e3−1.29e4i)T+(−6.75e8−1.16e9i)T2 1 + (7.49e3 - 1.29e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1−2.50e4T+1.80e9T2 1 - 2.50e4T + 1.80e9T^{2}
73 1+(−5.74e3+9.94e3i)T+(−1.03e9−1.79e9i)T2 1 + (-5.74e3 + 9.94e3i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(−2.70e4−4.67e4i)T+(−1.53e9+2.66e9i)T2 1 + (-2.70e4 - 4.67e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1−1.21e4T+3.93e9T2 1 - 1.21e4T + 3.93e9T^{2}
89 1+(−4.45e4−7.71e4i)T+(−2.79e9+4.83e9i)T2 1 + (-4.45e4 - 7.71e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1+3.58e4T+8.58e9T2 1 + 3.58e4T + 8.58e9T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−9.983434871023655906683557544157, −9.375801446252779900148752739949, −8.403010523582161637776921814206, −7.48604831012962188112347549751, −6.49158636155411251435873686893, −5.23641202956109717241233997441, −4.69844258414365679295064646166, −3.57746665008316057433151674246, −2.30861739763487503342694953904, −0.74983469601561607380404949101, 0.29162224823871025266661263449, 1.73735678507461767663967026368, 2.85050626624872558284038439762, 4.00593786637996127065493749911, 5.24048830953022228231909458603, 6.13326702175902869832949516158, 7.21185261103063538891272431401, 7.65380014309563899315013086958, 8.793290123791262103038151508258, 9.824257090554753333401471114441

Graph of the ZZ-function along the critical line