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Properties

Label 2-252-7.2-c7-0-8
Degree 22
Conductor 252252
Sign 0.974+0.224i0.974 + 0.224i
Analytic cond. 78.721078.7210
Root an. cond. 8.872488.87248
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−161. + 279. i)5-s + (−882. + 210. i)7-s + (43.6 + 75.6i)11-s − 9.60e3·13-s + (−1.61e4 − 2.80e4i)17-s + (−477. + 826. i)19-s + (4.78e3 − 8.27e3i)23-s + (−1.29e4 − 2.24e4i)25-s − 2.10e5·29-s + (2.08e4 + 3.60e4i)31-s + (8.36e4 − 2.80e5i)35-s + (−1.39e4 + 2.42e4i)37-s + 2.96e5·41-s + 4.96e5·43-s + (−3.08e5 + 5.34e5i)47-s + ⋯
L(s)  = 1  + (−0.576 + 0.999i)5-s + (−0.972 + 0.231i)7-s + (0.00989 + 0.0171i)11-s − 1.21·13-s + (−0.799 − 1.38i)17-s + (−0.0159 + 0.0276i)19-s + (0.0819 − 0.141i)23-s + (−0.165 − 0.286i)25-s − 1.59·29-s + (0.125 + 0.217i)31-s + (0.329 − 1.10i)35-s + (−0.0454 + 0.0786i)37-s + 0.671·41-s + 0.951·43-s + (−0.433 + 0.750i)47-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=((0.974+0.224i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+7/2) L(s)=((0.974+0.224i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: 0.974+0.224i0.974 + 0.224i
Analytic conductor: 78.721078.7210
Root analytic conductor: 8.872488.87248
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ252(37,⋅)\chi_{252} (37, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 252, ( :7/2), 0.974+0.224i)(2,\ 252,\ (\ :7/2),\ 0.974 + 0.224i)

Particular Values

L(4)L(4) ≈\approx 0.73902388940.7390238894
L(12)L(\frac12) ≈\approx 0.73902388940.7390238894
L(92)L(\frac{9}{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 1
7 1+(882.−210.i)T 1 + (882. - 210. i)T
good5 1+(161.−279.i)T+(−3.90e4−6.76e4i)T2 1 + (161. - 279. i)T + (-3.90e4 - 6.76e4i)T^{2}
11 1+(−43.6−75.6i)T+(−9.74e6+1.68e7i)T2 1 + (-43.6 - 75.6i)T + (-9.74e6 + 1.68e7i)T^{2}
13 1+9.60e3T+6.27e7T2 1 + 9.60e3T + 6.27e7T^{2}
17 1+(1.61e4+2.80e4i)T+(−2.05e8+3.55e8i)T2 1 + (1.61e4 + 2.80e4i)T + (-2.05e8 + 3.55e8i)T^{2}
19 1+(477.−826.i)T+(−4.46e8−7.74e8i)T2 1 + (477. - 826. i)T + (-4.46e8 - 7.74e8i)T^{2}
23 1+(−4.78e3+8.27e3i)T+(−1.70e9−2.94e9i)T2 1 + (-4.78e3 + 8.27e3i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+2.10e5T+1.72e10T2 1 + 2.10e5T + 1.72e10T^{2}
31 1+(−2.08e4−3.60e4i)T+(−1.37e10+2.38e10i)T2 1 + (-2.08e4 - 3.60e4i)T + (-1.37e10 + 2.38e10i)T^{2}
37 1+(1.39e4−2.42e4i)T+(−4.74e10−8.22e10i)T2 1 + (1.39e4 - 2.42e4i)T + (-4.74e10 - 8.22e10i)T^{2}
41 1−2.96e5T+1.94e11T2 1 - 2.96e5T + 1.94e11T^{2}
43 1−4.96e5T+2.71e11T2 1 - 4.96e5T + 2.71e11T^{2}
47 1+(3.08e5−5.34e5i)T+(−2.53e11−4.38e11i)T2 1 + (3.08e5 - 5.34e5i)T + (-2.53e11 - 4.38e11i)T^{2}
53 1+(−2.99e5−5.19e5i)T+(−5.87e11+1.01e12i)T2 1 + (-2.99e5 - 5.19e5i)T + (-5.87e11 + 1.01e12i)T^{2}
59 1+(−1.20e6−2.08e6i)T+(−1.24e12+2.15e12i)T2 1 + (-1.20e6 - 2.08e6i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(5.39e5−9.34e5i)T+(−1.57e12−2.72e12i)T2 1 + (5.39e5 - 9.34e5i)T + (-1.57e12 - 2.72e12i)T^{2}
67 1+(1.37e6+2.37e6i)T+(−3.03e12+5.24e12i)T2 1 + (1.37e6 + 2.37e6i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1+2.21e6T+9.09e12T2 1 + 2.21e6T + 9.09e12T^{2}
73 1+(−1.33e6−2.30e6i)T+(−5.52e12+9.56e12i)T2 1 + (-1.33e6 - 2.30e6i)T + (-5.52e12 + 9.56e12i)T^{2}
79 1+(−1.44e6+2.50e6i)T+(−9.60e12−1.66e13i)T2 1 + (-1.44e6 + 2.50e6i)T + (-9.60e12 - 1.66e13i)T^{2}
83 1+3.60e6T+2.71e13T2 1 + 3.60e6T + 2.71e13T^{2}
89 1+(−4.66e5+8.07e5i)T+(−2.21e13−3.83e13i)T2 1 + (-4.66e5 + 8.07e5i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1−1.21e7T+8.07e13T2 1 - 1.21e7T + 8.07e13T^{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

−10.82393456093027556845736392609, −9.762770946311741740387332219668, −9.039307733849699898011237798845, −7.40708834373099152771015067598, −7.06501737213909625970077115932, −5.82977140737949768784297445871, −4.45843800580939805678533071722, −3.16754878059358489046102373176, −2.42230629108170508282349946301, −0.31136013973892167230369183641, 0.55008896058878543023097529060, 2.10852900048182298097533728272, 3.64571675139886084670192580203, 4.52765910315232382230004733506, 5.74291686828240377826743588356, 6.93226322262324149877603523729, 7.936366282360208993011229512637, 8.956079319386969873990670471943, 9.743586596193758588435150465047, 10.81495923188686095418204021921

Graph of the ZZ-function along the critical line