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Properties

Label 2-252-21.5-c7-0-9
Degree 22
Conductor 252252
Sign 0.893+0.448i0.893 + 0.448i
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  + (116. − 202. i)5-s + (−84.3 − 903. i)7-s + (−778. + 449. i)11-s + 6.65e3i·13-s + (1.20e4 + 2.09e4i)17-s + (1.31e4 + 7.58e3i)19-s + (4.56e4 + 2.63e4i)23-s + (1.18e4 + 2.05e4i)25-s + 3.62e4i·29-s + (1.31e5 − 7.56e4i)31-s + (−1.92e5 − 8.83e4i)35-s + (3.06e4 − 5.30e4i)37-s − 2.66e5·41-s + 3.23e5·43-s + (3.52e5 − 6.09e5i)47-s + ⋯
L(s)  = 1  + (0.417 − 0.722i)5-s + (−0.0929 − 0.995i)7-s + (−0.176 + 0.101i)11-s + 0.840i·13-s + (0.596 + 1.03i)17-s + (0.439 + 0.253i)19-s + (0.782 + 0.451i)23-s + (0.151 + 0.262i)25-s + 0.275i·29-s + (0.790 − 0.456i)31-s + (−0.758 − 0.348i)35-s + (0.0994 − 0.172i)37-s − 0.602·41-s + 0.619·43-s + (0.494 − 0.856i)47-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=((0.893+0.448i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+7/2) L(s)=((0.893+0.448i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: 0.893+0.448i0.893 + 0.448i
Analytic conductor: 78.721078.7210
Root analytic conductor: 8.872488.87248
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ252(89,⋅)\chi_{252} (89, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 252, ( :7/2), 0.893+0.448i)(2,\ 252,\ (\ :7/2),\ 0.893 + 0.448i)

Particular Values

L(4)L(4) ≈\approx 2.4217077362.421707736
L(12)L(\frac12) ≈\approx 2.4217077362.421707736
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+(84.3+903.i)T 1 + (84.3 + 903. i)T
good5 1+(−116.+202.i)T+(−3.90e4−6.76e4i)T2 1 + (-116. + 202. i)T + (-3.90e4 - 6.76e4i)T^{2}
11 1+(778.−449.i)T+(9.74e6−1.68e7i)T2 1 + (778. - 449. i)T + (9.74e6 - 1.68e7i)T^{2}
13 1−6.65e3iT−6.27e7T2 1 - 6.65e3iT - 6.27e7T^{2}
17 1+(−1.20e4−2.09e4i)T+(−2.05e8+3.55e8i)T2 1 + (-1.20e4 - 2.09e4i)T + (-2.05e8 + 3.55e8i)T^{2}
19 1+(−1.31e4−7.58e3i)T+(4.46e8+7.74e8i)T2 1 + (-1.31e4 - 7.58e3i)T + (4.46e8 + 7.74e8i)T^{2}
23 1+(−4.56e4−2.63e4i)T+(1.70e9+2.94e9i)T2 1 + (-4.56e4 - 2.63e4i)T + (1.70e9 + 2.94e9i)T^{2}
29 1−3.62e4iT−1.72e10T2 1 - 3.62e4iT - 1.72e10T^{2}
31 1+(−1.31e5+7.56e4i)T+(1.37e10−2.38e10i)T2 1 + (-1.31e5 + 7.56e4i)T + (1.37e10 - 2.38e10i)T^{2}
37 1+(−3.06e4+5.30e4i)T+(−4.74e10−8.22e10i)T2 1 + (-3.06e4 + 5.30e4i)T + (-4.74e10 - 8.22e10i)T^{2}
41 1+2.66e5T+1.94e11T2 1 + 2.66e5T + 1.94e11T^{2}
43 1−3.23e5T+2.71e11T2 1 - 3.23e5T + 2.71e11T^{2}
47 1+(−3.52e5+6.09e5i)T+(−2.53e11−4.38e11i)T2 1 + (-3.52e5 + 6.09e5i)T + (-2.53e11 - 4.38e11i)T^{2}
53 1+(−6.24e5+3.60e5i)T+(5.87e11−1.01e12i)T2 1 + (-6.24e5 + 3.60e5i)T + (5.87e11 - 1.01e12i)T^{2}
59 1+(8.13e4+1.40e5i)T+(−1.24e12+2.15e12i)T2 1 + (8.13e4 + 1.40e5i)T + (-1.24e12 + 2.15e12i)T^{2}
61 1+(1.14e6+6.59e5i)T+(1.57e12+2.72e12i)T2 1 + (1.14e6 + 6.59e5i)T + (1.57e12 + 2.72e12i)T^{2}
67 1+(−1.74e6−3.01e6i)T+(−3.03e12+5.24e12i)T2 1 + (-1.74e6 - 3.01e6i)T + (-3.03e12 + 5.24e12i)T^{2}
71 1+3.93e6iT−9.09e12T2 1 + 3.93e6iT - 9.09e12T^{2}
73 1+(−1.29e6+7.45e5i)T+(5.52e12−9.56e12i)T2 1 + (-1.29e6 + 7.45e5i)T + (5.52e12 - 9.56e12i)T^{2}
79 1+(5.09e5−8.83e5i)T+(−9.60e12−1.66e13i)T2 1 + (5.09e5 - 8.83e5i)T + (-9.60e12 - 1.66e13i)T^{2}
83 1+1.08e5T+2.71e13T2 1 + 1.08e5T + 2.71e13T^{2}
89 1+(−4.21e6+7.30e6i)T+(−2.21e13−3.83e13i)T2 1 + (-4.21e6 + 7.30e6i)T + (-2.21e13 - 3.83e13i)T^{2}
97 1+3.46e6iT−8.07e13T2 1 + 3.46e6iT - 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

−10.61060088055948102611799106291, −9.780829879875516391662480342249, −8.872158057187969162535914509445, −7.77380229757933012894364318187, −6.79034836218760961934260435843, −5.59341809759198064404040237858, −4.52531887230968603217056251814, −3.45940327895202494053923148318, −1.74764362884211710937530660187, −0.815627264133188656061681878104, 0.813885557592802965463263884743, 2.52355683956879776850884567225, 3.08579742006466132916765471644, 4.92565554462580401884882228345, 5.81769380681922821888498914134, 6.82304717772476062690606048032, 7.936002453567447534300755259444, 9.023412349482066246891972795643, 9.935902218435212212272371100157, 10.80343949842778124034256872832

Graph of the ZZ-function along the critical line