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Properties

Label 2-20e2-100.23-c1-0-4
Degree 22
Conductor 400400
Sign 0.464−0.885i0.464 - 0.885i
Analytic cond. 3.194013.19401
Root an. cond. 1.787181.78718
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.13 + 2.21i)3-s + (−0.139 − 2.23i)5-s + (−0.125 − 0.125i)7-s + (−1.88 + 2.59i)9-s + (3.73 + 5.14i)11-s + (0.413 + 2.61i)13-s + (4.79 − 2.83i)15-s + (3.67 + 1.87i)17-s + (−0.529 − 1.62i)19-s + (0.136 − 0.421i)21-s + (0.203 − 1.28i)23-s + (−4.96 + 0.621i)25-s + (−0.501 − 0.0794i)27-s + (−8.12 − 2.64i)29-s + (1.69 − 0.551i)31-s + ⋯
L(s)  = 1  + (0.652 + 1.28i)3-s + (−0.0622 − 0.998i)5-s + (−0.0475 − 0.0475i)7-s + (−0.627 + 0.864i)9-s + (1.12 + 1.55i)11-s + (0.114 + 0.723i)13-s + (1.23 − 0.731i)15-s + (0.891 + 0.454i)17-s + (−0.121 − 0.373i)19-s + (0.0298 − 0.0919i)21-s + (0.0425 − 0.268i)23-s + (−0.992 + 0.124i)25-s + (−0.0965 − 0.0152i)27-s + (−1.50 − 0.490i)29-s + (0.304 − 0.0989i)31-s + ⋯

Functional equation

Λ(s)=(400s/2 ΓC(s) L(s)=((0.464−0.885i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(400s/2 ΓC(s+1/2) L(s)=((0.464−0.885i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 400400    =    24⋅522^{4} \cdot 5^{2}
Sign: 0.464−0.885i0.464 - 0.885i
Analytic conductor: 3.194013.19401
Root analytic conductor: 1.787181.78718
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ400(223,⋅)\chi_{400} (223, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 400, ( :1/2), 0.464−0.885i)(2,\ 400,\ (\ :1/2),\ 0.464 - 0.885i)

Particular Values

L(1)L(1) ≈\approx 1.47827+0.894149i1.47827 + 0.894149i
L(12)L(\frac12) ≈\approx 1.47827+0.894149i1.47827 + 0.894149i
L(32)L(\frac{3}{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
5 1+(0.139+2.23i)T 1 + (0.139 + 2.23i)T
good3 1+(−1.13−2.21i)T+(−1.76+2.42i)T2 1 + (-1.13 - 2.21i)T + (-1.76 + 2.42i)T^{2}
7 1+(0.125+0.125i)T+7iT2 1 + (0.125 + 0.125i)T + 7iT^{2}
11 1+(−3.73−5.14i)T+(−3.39+10.4i)T2 1 + (-3.73 - 5.14i)T + (-3.39 + 10.4i)T^{2}
13 1+(−0.413−2.61i)T+(−12.3+4.01i)T2 1 + (-0.413 - 2.61i)T + (-12.3 + 4.01i)T^{2}
17 1+(−3.67−1.87i)T+(9.99+13.7i)T2 1 + (-3.67 - 1.87i)T + (9.99 + 13.7i)T^{2}
19 1+(0.529+1.62i)T+(−15.3+11.1i)T2 1 + (0.529 + 1.62i)T + (-15.3 + 11.1i)T^{2}
23 1+(−0.203+1.28i)T+(−21.8−7.10i)T2 1 + (-0.203 + 1.28i)T + (-21.8 - 7.10i)T^{2}
29 1+(8.12+2.64i)T+(23.4+17.0i)T2 1 + (8.12 + 2.64i)T + (23.4 + 17.0i)T^{2}
31 1+(−1.69+0.551i)T+(25.0−18.2i)T2 1 + (-1.69 + 0.551i)T + (25.0 - 18.2i)T^{2}
37 1+(−6.42+1.01i)T+(35.1−11.4i)T2 1 + (-6.42 + 1.01i)T + (35.1 - 11.4i)T^{2}
41 1+(8.51+6.18i)T+(12.6+38.9i)T2 1 + (8.51 + 6.18i)T + (12.6 + 38.9i)T^{2}
43 1+(−2.70+2.70i)T−43iT2 1 + (-2.70 + 2.70i)T - 43iT^{2}
47 1+(9.15−4.66i)T+(27.6−38.0i)T2 1 + (9.15 - 4.66i)T + (27.6 - 38.0i)T^{2}
53 1+(−8.64+4.40i)T+(31.1−42.8i)T2 1 + (-8.64 + 4.40i)T + (31.1 - 42.8i)T^{2}
59 1+(4.80+3.49i)T+(18.2+56.1i)T2 1 + (4.80 + 3.49i)T + (18.2 + 56.1i)T^{2}
61 1+(−6.89+5.00i)T+(18.8−58.0i)T2 1 + (-6.89 + 5.00i)T + (18.8 - 58.0i)T^{2}
67 1+(−1.50+2.95i)T+(−39.3−54.2i)T2 1 + (-1.50 + 2.95i)T + (-39.3 - 54.2i)T^{2}
71 1+(7.64+2.48i)T+(57.4+41.7i)T2 1 + (7.64 + 2.48i)T + (57.4 + 41.7i)T^{2}
73 1+(1.62+0.256i)T+(69.4+22.5i)T2 1 + (1.62 + 0.256i)T + (69.4 + 22.5i)T^{2}
79 1+(2.78−8.57i)T+(−63.9−46.4i)T2 1 + (2.78 - 8.57i)T + (-63.9 - 46.4i)T^{2}
83 1+(−9.56−4.87i)T+(48.7+67.1i)T2 1 + (-9.56 - 4.87i)T + (48.7 + 67.1i)T^{2}
89 1+(−0.221−0.305i)T+(−27.5+84.6i)T2 1 + (-0.221 - 0.305i)T + (-27.5 + 84.6i)T^{2}
97 1+(4.08+8.01i)T+(−57.0+78.4i)T2 1 + (4.08 + 8.01i)T + (-57.0 + 78.4i)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

−11.46006234961153961223927925060, −10.12361426433823864362831215960, −9.525337282439968383131728173637, −9.018058129993675392080446048272, −8.010636587256027396938444044347, −6.76877734014291368313075862314, −5.27319590693219223508566239915, −4.30889185830118886641707642789, −3.77897433679903852536967966752, −1.86300820522912989755183607168, 1.26905290318139920187791479281, 2.86876940834873235912507460317, 3.57327593618841218177305262031, 5.76685573878648905883480925399, 6.48454814624975689077934845401, 7.46030594318474630822653403105, 8.123275276621929058972356316380, 9.086322259266826826447003219311, 10.25184931383818879730310026285, 11.36143867415608032678139527670

Graph of the ZZ-function along the critical line