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Properties

Label 2-588-7.4-c5-0-22
Degree 22
Conductor 588588
Sign 0.605+0.795i0.605 + 0.795i
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 + (39.3 + 68.1i)5-s + (−40.5 − 70.1i)9-s + (−345. + 598. i)11-s − 818.·13-s − 708.·15-s + (554. − 960. i)17-s + (−286. − 496. i)19-s + (−1.25e3 − 2.18e3i)23-s + (−1.53e3 + 2.65e3i)25-s + 729·27-s − 3.25e3·29-s + (5.05e3 − 8.76e3i)31-s + (−3.11e3 − 5.38e3i)33-s + (−2.43e3 − 4.21e3i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.703 + 1.21i)5-s + (−0.166 − 0.288i)9-s + (−0.861 + 1.49i)11-s − 1.34·13-s − 0.812·15-s + (0.465 − 0.806i)17-s + (−0.182 − 0.315i)19-s + (−0.496 − 0.859i)23-s + (−0.490 + 0.849i)25-s + 0.192·27-s − 0.719·29-s + (0.945 − 1.63i)31-s + (−0.497 − 0.861i)33-s + (−0.292 − 0.506i)37-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=((0.605+0.795i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+5/2) L(s)=((0.605+0.795i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: 0.605+0.795i0.605 + 0.795i
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), 0.605+0.795i)(2,\ 588,\ (\ :5/2),\ 0.605 + 0.795i)

Particular Values

L(3)L(3) ≈\approx 0.65196884960.6519688496
L(12)L(\frac12) ≈\approx 0.65196884960.6519688496
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+(−39.3−68.1i)T+(−1.56e3+2.70e3i)T2 1 + (-39.3 - 68.1i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(345.−598.i)T+(−8.05e4−1.39e5i)T2 1 + (345. - 598. i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1+818.T+3.71e5T2 1 + 818.T + 3.71e5T^{2}
17 1+(−554.+960.i)T+(−7.09e5−1.22e6i)T2 1 + (-554. + 960. i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(286.+496.i)T+(−1.23e6+2.14e6i)T2 1 + (286. + 496. i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(1.25e3+2.18e3i)T+(−3.21e6+5.57e6i)T2 1 + (1.25e3 + 2.18e3i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1+3.25e3T+2.05e7T2 1 + 3.25e3T + 2.05e7T^{2}
31 1+(−5.05e3+8.76e3i)T+(−1.43e7−2.47e7i)T2 1 + (-5.05e3 + 8.76e3i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(2.43e3+4.21e3i)T+(−3.46e7+6.00e7i)T2 1 + (2.43e3 + 4.21e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1−1.30e4T+1.15e8T2 1 - 1.30e4T + 1.15e8T^{2}
43 1+9.30e3T+1.47e8T2 1 + 9.30e3T + 1.47e8T^{2}
47 1+(−6.45e3−1.11e4i)T+(−1.14e8+1.98e8i)T2 1 + (-6.45e3 - 1.11e4i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(−9.77e3+1.69e4i)T+(−2.09e8−3.62e8i)T2 1 + (-9.77e3 + 1.69e4i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(1.25e4−2.17e4i)T+(−3.57e8−6.19e8i)T2 1 + (1.25e4 - 2.17e4i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(1.56e4+2.71e4i)T+(−4.22e8+7.31e8i)T2 1 + (1.56e4 + 2.71e4i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(2.79e4−4.84e4i)T+(−6.75e8−1.16e9i)T2 1 + (2.79e4 - 4.84e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1−2.05e4T+1.80e9T2 1 - 2.05e4T + 1.80e9T^{2}
73 1+(3.38e4−5.85e4i)T+(−1.03e9−1.79e9i)T2 1 + (3.38e4 - 5.85e4i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(7.03e3+1.21e4i)T+(−1.53e9+2.66e9i)T2 1 + (7.03e3 + 1.21e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1−7.71e4T+3.93e9T2 1 - 7.71e4T + 3.93e9T^{2}
89 1+(−160.−277.i)T+(−2.79e9+4.83e9i)T2 1 + (-160. - 277. i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1−1.12e5T+8.58e9T2 1 - 1.12e5T + 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

−9.998486741416113647930281721139, −9.347570929574025191321589420179, −7.69390171999171770428906054037, −7.15591689020391763104503948984, −6.14733292967741216363395231288, −5.13955098848657169366369657713, −4.29459992638884569441479682175, −2.68181713973450480556465538386, −2.27216269510107732894365325685, −0.16350537271178732751894324320, 0.937635321912667336934340148633, 1.93353275365786013877437808135, 3.22921224745443967011301324273, 4.79614377797507738807053062380, 5.51434185144685156469246146685, 6.15341726245543779962908989374, 7.54642164592174511153403678563, 8.277784635628010683501585053747, 9.069166137143647470921573014543, 10.07063365993961363091190051477

Graph of the ZZ-function along the critical line