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Properties

Label 2-252-21.17-c7-0-3
Degree 22
Conductor 252252
Sign 0.888−0.458i0.888 - 0.458i
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  + (−275. − 477. i)5-s + (877. − 230. i)7-s + (1.25e3 + 722. i)11-s + 1.24e4i·13-s + (8.65e3 − 1.49e4i)17-s + (−2.25e4 + 1.30e4i)19-s + (−6.79e4 + 3.92e4i)23-s + (−1.12e5 + 1.95e5i)25-s − 3.35e4i·29-s + (1.96e5 + 1.13e5i)31-s + (−3.52e5 − 3.55e5i)35-s + (−1.75e5 − 3.03e5i)37-s + 1.09e5·41-s − 1.10e5·43-s + (5.13e5 + 8.90e5i)47-s + ⋯
L(s)  = 1  + (−0.985 − 1.70i)5-s + (0.967 − 0.254i)7-s + (0.283 + 0.163i)11-s + 1.57i·13-s + (0.427 − 0.740i)17-s + (−0.754 + 0.435i)19-s + (−1.16 + 0.672i)23-s + (−1.44 + 2.50i)25-s − 0.255i·29-s + (1.18 + 0.683i)31-s + (−1.38 − 1.40i)35-s + (−0.568 − 0.984i)37-s + 0.248·41-s − 0.211·43-s + (0.722 + 1.25i)47-s + ⋯

Functional equation

Λ(s)=(252s/2 ΓC(s) L(s)=((0.888−0.458i) Λ‾(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(8-s) \end{aligned}
Λ(s)=(252s/2 ΓC(s+7/2) L(s)=((0.888−0.458i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 252252    =    22⋅32⋅72^{2} \cdot 3^{2} \cdot 7
Sign: 0.888−0.458i0.888 - 0.458i
Analytic conductor: 78.721078.7210
Root analytic conductor: 8.872488.87248
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: χ252(17,⋅)\chi_{252} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 252, ( :7/2), 0.888−0.458i)(2,\ 252,\ (\ :7/2),\ 0.888 - 0.458i)

Particular Values

L(4)L(4) ≈\approx 1.4624843991.462484399
L(12)L(\frac12) ≈\approx 1.4624843991.462484399
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+(−877.+230.i)T 1 + (-877. + 230. i)T
good5 1+(275.+477.i)T+(−3.90e4+6.76e4i)T2 1 + (275. + 477. i)T + (-3.90e4 + 6.76e4i)T^{2}
11 1+(−1.25e3−722.i)T+(9.74e6+1.68e7i)T2 1 + (-1.25e3 - 722. i)T + (9.74e6 + 1.68e7i)T^{2}
13 1−1.24e4iT−6.27e7T2 1 - 1.24e4iT - 6.27e7T^{2}
17 1+(−8.65e3+1.49e4i)T+(−2.05e8−3.55e8i)T2 1 + (-8.65e3 + 1.49e4i)T + (-2.05e8 - 3.55e8i)T^{2}
19 1+(2.25e4−1.30e4i)T+(4.46e8−7.74e8i)T2 1 + (2.25e4 - 1.30e4i)T + (4.46e8 - 7.74e8i)T^{2}
23 1+(6.79e4−3.92e4i)T+(1.70e9−2.94e9i)T2 1 + (6.79e4 - 3.92e4i)T + (1.70e9 - 2.94e9i)T^{2}
29 1+3.35e4iT−1.72e10T2 1 + 3.35e4iT - 1.72e10T^{2}
31 1+(−1.96e5−1.13e5i)T+(1.37e10+2.38e10i)T2 1 + (-1.96e5 - 1.13e5i)T + (1.37e10 + 2.38e10i)T^{2}
37 1+(1.75e5+3.03e5i)T+(−4.74e10+8.22e10i)T2 1 + (1.75e5 + 3.03e5i)T + (-4.74e10 + 8.22e10i)T^{2}
41 1−1.09e5T+1.94e11T2 1 - 1.09e5T + 1.94e11T^{2}
43 1+1.10e5T+2.71e11T2 1 + 1.10e5T + 2.71e11T^{2}
47 1+(−5.13e5−8.90e5i)T+(−2.53e11+4.38e11i)T2 1 + (-5.13e5 - 8.90e5i)T + (-2.53e11 + 4.38e11i)T^{2}
53 1+(−8.68e5−5.01e5i)T+(5.87e11+1.01e12i)T2 1 + (-8.68e5 - 5.01e5i)T + (5.87e11 + 1.01e12i)T^{2}
59 1+(1.13e5−1.95e5i)T+(−1.24e12−2.15e12i)T2 1 + (1.13e5 - 1.95e5i)T + (-1.24e12 - 2.15e12i)T^{2}
61 1+(−3.40e5+1.96e5i)T+(1.57e12−2.72e12i)T2 1 + (-3.40e5 + 1.96e5i)T + (1.57e12 - 2.72e12i)T^{2}
67 1+(3.93e5−6.81e5i)T+(−3.03e12−5.24e12i)T2 1 + (3.93e5 - 6.81e5i)T + (-3.03e12 - 5.24e12i)T^{2}
71 1−2.32e6iT−9.09e12T2 1 - 2.32e6iT - 9.09e12T^{2}
73 1+(−1.33e6−7.71e5i)T+(5.52e12+9.56e12i)T2 1 + (-1.33e6 - 7.71e5i)T + (5.52e12 + 9.56e12i)T^{2}
79 1+(−3.43e6−5.95e6i)T+(−9.60e12+1.66e13i)T2 1 + (-3.43e6 - 5.95e6i)T + (-9.60e12 + 1.66e13i)T^{2}
83 1+6.19e6T+2.71e13T2 1 + 6.19e6T + 2.71e13T^{2}
89 1+(4.21e6+7.29e6i)T+(−2.21e13+3.83e13i)T2 1 + (4.21e6 + 7.29e6i)T + (-2.21e13 + 3.83e13i)T^{2}
97 1−7.20e6iT−8.07e13T2 1 - 7.20e6iT - 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

−11.19520849858502517032775046567, −9.676752529181927530883175933515, −8.785454839525135595945938433133, −8.090799632316520467142286945360, −7.17289380611051265225240266463, −5.54883142114836322484415375681, −4.43117870047202521226013095372, −4.07220059943959991860344210315, −1.82327209820503143712834361654, −0.934087369537489235001450806867, 0.41950839292759119189691664231, 2.25189105018520391432076749621, 3.27658451381418353450624255398, 4.30529844128973499830518323978, 5.82761225807155728546668295170, 6.80352795388277502565551097492, 7.973706296411023390611856818983, 8.288845843152609582668395510009, 10.29162214787882992001608773720, 10.60363475328915229077830011532

Graph of the ZZ-function along the critical line