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Properties

Label 2-588-7.2-c5-0-15
Degree 22
Conductor 588588
Sign 0.991+0.126i0.991 + 0.126i
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 + (14.2 − 24.6i)5-s + (−40.5 + 70.1i)9-s + (−212. − 367. i)11-s + 508.·13-s − 255.·15-s + (269. + 467. i)17-s + (−1.30e3 + 2.25e3i)19-s + (−130. + 226. i)23-s + (1.15e3 + 2.00e3i)25-s + 729·27-s + 6.87e3·29-s + (−2.84e3 − 4.92e3i)31-s + (−1.90e3 + 3.30e3i)33-s + (−2.45e3 + 4.25e3i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.254 − 0.440i)5-s + (−0.166 + 0.288i)9-s + (−0.528 − 0.915i)11-s + 0.834·13-s − 0.293·15-s + (0.226 + 0.392i)17-s + (−0.827 + 1.43i)19-s + (−0.0514 + 0.0891i)23-s + (0.370 + 0.642i)25-s + 0.192·27-s + 1.51·29-s + (−0.531 − 0.920i)31-s + (−0.305 + 0.528i)33-s + (−0.294 + 0.510i)37-s + ⋯

Functional equation

Λ(s)=(588s/2 ΓC(s) L(s)=((0.991+0.126i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(588s/2 ΓC(s+5/2) L(s)=((0.991+0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 588588    =    22⋅3⋅722^{2} \cdot 3 \cdot 7^{2}
Sign: 0.991+0.126i0.991 + 0.126i
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), 0.991+0.126i)(2,\ 588,\ (\ :5/2),\ 0.991 + 0.126i)

Particular Values

L(3)L(3) ≈\approx 1.8200827441.820082744
L(12)L(\frac12) ≈\approx 1.8200827441.820082744
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+(−14.2+24.6i)T+(−1.56e3−2.70e3i)T2 1 + (-14.2 + 24.6i)T + (-1.56e3 - 2.70e3i)T^{2}
11 1+(212.+367.i)T+(−8.05e4+1.39e5i)T2 1 + (212. + 367. i)T + (-8.05e4 + 1.39e5i)T^{2}
13 1−508.T+3.71e5T2 1 - 508.T + 3.71e5T^{2}
17 1+(−269.−467.i)T+(−7.09e5+1.22e6i)T2 1 + (-269. - 467. i)T + (-7.09e5 + 1.22e6i)T^{2}
19 1+(1.30e3−2.25e3i)T+(−1.23e6−2.14e6i)T2 1 + (1.30e3 - 2.25e3i)T + (-1.23e6 - 2.14e6i)T^{2}
23 1+(130.−226.i)T+(−3.21e6−5.57e6i)T2 1 + (130. - 226. i)T + (-3.21e6 - 5.57e6i)T^{2}
29 1−6.87e3T+2.05e7T2 1 - 6.87e3T + 2.05e7T^{2}
31 1+(2.84e3+4.92e3i)T+(−1.43e7+2.47e7i)T2 1 + (2.84e3 + 4.92e3i)T + (-1.43e7 + 2.47e7i)T^{2}
37 1+(2.45e3−4.25e3i)T+(−3.46e7−6.00e7i)T2 1 + (2.45e3 - 4.25e3i)T + (-3.46e7 - 6.00e7i)T^{2}
41 1+5.72e3T+1.15e8T2 1 + 5.72e3T + 1.15e8T^{2}
43 1+1.73e3T+1.47e8T2 1 + 1.73e3T + 1.47e8T^{2}
47 1+(−5.07e3+8.78e3i)T+(−1.14e8−1.98e8i)T2 1 + (-5.07e3 + 8.78e3i)T + (-1.14e8 - 1.98e8i)T^{2}
53 1+(−1.55e4−2.70e4i)T+(−2.09e8+3.62e8i)T2 1 + (-1.55e4 - 2.70e4i)T + (-2.09e8 + 3.62e8i)T^{2}
59 1+(−1.94e4−3.36e4i)T+(−3.57e8+6.19e8i)T2 1 + (-1.94e4 - 3.36e4i)T + (-3.57e8 + 6.19e8i)T^{2}
61 1+(6.82e3−1.18e4i)T+(−4.22e8−7.31e8i)T2 1 + (6.82e3 - 1.18e4i)T + (-4.22e8 - 7.31e8i)T^{2}
67 1+(−1.53e4−2.66e4i)T+(−6.75e8+1.16e9i)T2 1 + (-1.53e4 - 2.66e4i)T + (-6.75e8 + 1.16e9i)T^{2}
71 1+4.56e4T+1.80e9T2 1 + 4.56e4T + 1.80e9T^{2}
73 1+(1.08e4+1.88e4i)T+(−1.03e9+1.79e9i)T2 1 + (1.08e4 + 1.88e4i)T + (-1.03e9 + 1.79e9i)T^{2}
79 1+(1.61e4−2.79e4i)T+(−1.53e9−2.66e9i)T2 1 + (1.61e4 - 2.79e4i)T + (-1.53e9 - 2.66e9i)T^{2}
83 1+4.66e4T+3.93e9T2 1 + 4.66e4T + 3.93e9T^{2}
89 1+(−3.18e4+5.52e4i)T+(−2.79e9−4.83e9i)T2 1 + (-3.18e4 + 5.52e4i)T + (-2.79e9 - 4.83e9i)T^{2}
97 1−1.15e5T+8.58e9T2 1 - 1.15e5T + 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

−10.09946153411825530877561109034, −8.658004138272186411748064485891, −8.361999266497959746342997473491, −7.21734164641702694942047517564, −6.00483429611358479369870261313, −5.66241305276809459412793061647, −4.28878859519690641187778836905, −3.12551797117269570882358658605, −1.74252898555037720374066540637, −0.799949472018642118560549603030, 0.56606536666878688454180978338, 2.15064019129525329310418272736, 3.18540312261477354782218810740, 4.46893756428488041674887058583, 5.18597431620250370995910690607, 6.43347697060738350784100460416, 7.01476631882102888865752676077, 8.318144651664217420308685263858, 9.106243368692240533309244670116, 10.16568717877110038269241668142

Graph of the ZZ-function along the critical line