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Properties

Label 2-588-7.2-c5-0-5
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 + (−15.9 + 27.6i)5-s + (−40.5 + 70.1i)9-s + (−130. − 225. i)11-s − 769.·13-s + 287.·15-s + (−776. − 1.34e3i)17-s + (375. − 649. i)19-s + (−377. + 653. i)23-s + (1.05e3 + 1.82e3i)25-s + 729·27-s + 6.00e3·29-s + (−3.21e3 − 5.55e3i)31-s + (−1.17e3 + 2.03e3i)33-s + (2.38e3 − 4.13e3i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.285 + 0.495i)5-s + (−0.166 + 0.288i)9-s + (−0.325 − 0.562i)11-s − 1.26·13-s + 0.330·15-s + (−0.651 − 1.12i)17-s + (0.238 − 0.412i)19-s + (−0.148 + 0.257i)23-s + (0.336 + 0.582i)25-s + 0.192·27-s + 1.32·29-s + (−0.599 − 1.03i)31-s + (−0.187 + 0.325i)33-s + (0.286 − 0.496i)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(373,⋅)\chi_{588} (373, \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 0.81380801830.8138080183
L(12)L(\frac12) ≈\approx 0.81380801830.8138080183
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+(15.9−27.6i)T+(−1.56e3−2.70e3i)T2 1 + (15.9 - 27.6i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(130.+225.i)T+(−8.05e4+1.39e5i)T2 1 + (130. + 225. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1+769.T+3.71e5T2 1 + 769.T + 3.71e5T^{2}
17 1+(776.+1.34e3i)T+(−7.09e5+1.22e6i)T2 1 + (776. + 1.34e3i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(−375.+649.i)T+(−1.23e6−2.14e6i)T2 1 + (-375. + 649. i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(377.−653.i)T+(−3.21e6−5.57e6i)T2 1 + (377. - 653. i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1−6.00e3T+2.05e7T2 1 - 6.00e3T + 2.05e7T^{2}
31 1+(3.21e3+5.55e3i)T+(−1.43e7+2.47e7i)T2 1 + (3.21e3 + 5.55e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(−2.38e3+4.13e3i)T+(−3.46e7−6.00e7i)T2 1 + (-2.38e3 + 4.13e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1−5.42e3T+1.15e8T2 1 - 5.42e3T + 1.15e8T^{2}
43 1+1.18e4T+1.47e8T2 1 + 1.18e4T + 1.47e8T^{2}
47 1+(8.71e3−1.50e4i)T+(−1.14e8−1.98e8i)T2 1 + (8.71e3 - 1.50e4i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(1.88e4+3.26e4i)T+(−2.09e8+3.62e8i)T2 1 + (1.88e4 + 3.26e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(−1.10e4−1.91e4i)T+(−3.57e8+6.19e8i)T2 1 + (-1.10e4 - 1.91e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(4.08e3−7.07e3i)T+(−4.22e8−7.31e8i)T2 1 + (4.08e3 - 7.07e3i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(6.50e3+1.12e4i)T+(−6.75e8+1.16e9i)T2 1 + (6.50e3 + 1.12e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1+1.23e4T+1.80e9T2 1 + 1.23e4T + 1.80e9T^{2}
73 1+(−2.18e4−3.77e4i)T+(−1.03e9+1.79e9i)T2 1 + (-2.18e4 - 3.77e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(3.83e4−6.64e4i)T+(−1.53e9−2.66e9i)T2 1 + (3.83e4 - 6.64e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1+2.18e4T+3.93e9T2 1 + 2.18e4T + 3.93e9T^{2}
89 1+(6.84e4−1.18e5i)T+(−2.79e9−4.83e9i)T2 1 + (6.84e4 - 1.18e5i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1−9.30e4T+8.58e9T2 1 - 9.30e4T + 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

−10.04863039081138182279377917370, −9.231003961595413095069555566307, −8.072530210998777706494762404070, −7.28567970380799604998258816587, −6.64372872747762649061109860804, −5.45420599643476119715017675012, −4.60423421140048572668946027780, −3.11286854645197789354337787934, −2.30236535280910748664620517995, −0.70568049398851539419256735121, 0.26767343996333052369294636123, 1.78406034225996911245922485760, 3.09511539833759666465238796610, 4.46232482733617553076625153097, 4.86497174274568538372822572289, 6.09798772785188995060767031164, 7.09226250658720736496133429294, 8.149229951522144959199136771036, 8.884296405336001080684895419057, 10.01830358340764411590627262832

Graph of the ZZ-function along the critical line