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Properties

Label 2-252-9.4-c7-0-24
Degree 22
Conductor 252252
Sign 0.915−0.401i0.915 - 0.401i
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  + (−6.63 + 46.2i)3-s + (117. − 203. i)5-s + (171.5 + 297. i)7-s + (−2.09e3 − 614. i)9-s + (1.30e3 + 2.25e3i)11-s + (3.24e3 − 5.61e3i)13-s + (8.65e3 + 6.80e3i)15-s − 1.28e4·17-s − 1.16e4·19-s + (−1.48e4 + 5.96e3i)21-s + (1.16e4 − 2.01e4i)23-s + (1.13e4 + 1.96e4i)25-s + (4.23e4 − 9.30e4i)27-s + (7.33e4 + 1.27e5i)29-s + (2.68e4 − 4.65e4i)31-s + ⋯
L(s)  = 1  + (−0.141 + 0.989i)3-s + (0.421 − 0.729i)5-s + (0.188 + 0.327i)7-s + (−0.959 − 0.280i)9-s + (0.295 + 0.511i)11-s + (0.409 − 0.709i)13-s + (0.662 + 0.520i)15-s − 0.634·17-s − 0.389·19-s + (−0.350 + 0.140i)21-s + (0.199 − 0.345i)23-s + (0.145 + 0.251i)25-s + (0.414 − 0.910i)27-s + (0.558 + 0.967i)29-s + (0.161 − 0.280i)31-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=((0.915−0.401i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+7/2) L(s)=((0.915−0.401i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: 0.915−0.401i0.915 - 0.401i
Analytic conductor: 78.721078.7210
Root analytic conductor: 8.872488.87248
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ252(85,⋅)\chi_{252} (85, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 252, ( :7/2), 0.915−0.401i)(2,\ 252,\ (\ :7/2),\ 0.915 - 0.401i)

Particular Values

L(4)L(4) ≈\approx 2.1578658002.157865800
L(12)L(\frac12) ≈\approx 2.1578658002.157865800
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+(6.63−46.2i)T 1 + (6.63 - 46.2i)T
7 1+(−171.5−297.i)T 1 + (-171.5 - 297. i)T
good5 1+(−117.+203.i)T+(−3.90e4−6.76e4i)T2 1 + (-117. + 203. i)T + (-3.90e4 - 6.76e4i)T^{2}
11 1+(−1.30e3−2.25e3i)T+(−9.74e6+1.68e7i)T2 1 + (-1.30e3 - 2.25e3i)T + (-9.74e6 + 1.68e7i)T^{2}
13 1+(−3.24e3+5.61e3i)T+(−3.13e7−5.43e7i)T2 1 + (-3.24e3 + 5.61e3i)T + (-3.13e7 - 5.43e7i)T^{2}
17 1+1.28e4T+4.10e8T2 1 + 1.28e4T + 4.10e8T^{2}
19 1+1.16e4T+8.93e8T2 1 + 1.16e4T + 8.93e8T^{2}
23 1+(−1.16e4+2.01e4i)T+(−1.70e9−2.94e9i)T2 1 + (-1.16e4 + 2.01e4i)T + (-1.70e9 - 2.94e9i)T^{2}
29 1+(−7.33e4−1.27e5i)T+(−8.62e9+1.49e10i)T2 1 + (-7.33e4 - 1.27e5i)T + (-8.62e9 + 1.49e10i)T^{2}
31 1+(−2.68e4+4.65e4i)T+(−1.37e10−2.38e10i)T2 1 + (-2.68e4 + 4.65e4i)T + (-1.37e10 - 2.38e10i)T^{2}
37 1−3.93e5T+9.49e10T2 1 - 3.93e5T + 9.49e10T^{2}
41 1+(−9.24e4+1.60e5i)T+(−9.73e10−1.68e11i)T2 1 + (-9.24e4 + 1.60e5i)T + (-9.73e10 - 1.68e11i)T^{2}
43 1+(4.49e5+7.79e5i)T+(−1.35e11+2.35e11i)T2 1 + (4.49e5 + 7.79e5i)T + (-1.35e11 + 2.35e11i)T^{2}
47 1+(6.77e5+1.17e6i)T+(−2.53e11+4.38e11i)T2 1 + (6.77e5 + 1.17e6i)T + (-2.53e11 + 4.38e11i)T^{2}
53 1−7.47e5T+1.17e12T2 1 - 7.47e5T + 1.17e12T^{2}
59 1+(−5.98e5+1.03e6i)T+(−1.24e12−2.15e12i)T2 1 + (-5.98e5 + 1.03e6i)T + (-1.24e12 - 2.15e12i)T^{2}
61 1+(−7.10e5−1.23e6i)T+(−1.57e12+2.72e12i)T2 1 + (-7.10e5 - 1.23e6i)T + (-1.57e12 + 2.72e12i)T^{2}
67 1+(1.46e6−2.52e6i)T+(−3.03e12−5.24e12i)T2 1 + (1.46e6 - 2.52e6i)T + (-3.03e12 - 5.24e12i)T^{2}
71 1−5.67e6T+9.09e12T2 1 - 5.67e6T + 9.09e12T^{2}
73 1−1.55e6T+1.10e13T2 1 - 1.55e6T + 1.10e13T^{2}
79 1+(−9.67e5−1.67e6i)T+(−9.60e12+1.66e13i)T2 1 + (-9.67e5 - 1.67e6i)T + (-9.60e12 + 1.66e13i)T^{2}
83 1+(−1.23e6−2.13e6i)T+(−1.35e13+2.35e13i)T2 1 + (-1.23e6 - 2.13e6i)T + (-1.35e13 + 2.35e13i)T^{2}
89 1−4.49e5T+4.42e13T2 1 - 4.49e5T + 4.42e13T^{2}
97 1+(−6.30e6−1.09e7i)T+(−4.03e13+6.99e13i)T2 1 + (-6.30e6 - 1.09e7i)T + (-4.03e13 + 6.99e13i)T^{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

−10.73583759208090016035573508919, −9.892576005728651440662810467887, −8.939052818203683283865023473200, −8.367046021630539759924778041083, −6.69699384813969485270666744034, −5.50337305500845956240096476426, −4.81377876574247419762040249636, −3.67397868459114955936956321557, −2.24176748548304434723663098444, −0.71808875503342142716856725453, 0.797991261793430309902219993206, 1.95880943279229773755380300910, 3.04169021787171460023262833684, 4.56680087698500387586217558363, 6.22480850964127799260137143471, 6.48993139445808244939826914492, 7.70558063999804806556605446012, 8.630776341464710253073165916029, 9.808209437174723290848718453746, 11.10434367338407785928399401699

Graph of the ZZ-function along the critical line