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Properties

Label 2-4000-125.103-c0-0-0
Degree 22
Conductor 40004000
Sign 0.0439−0.999i0.0439 - 0.999i
Analytic cond. 1.996261.99626
Root an. cond. 1.412891.41289
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.904 − 0.425i)5-s + (−0.770 + 0.637i)9-s + (−0.180 + 1.91i)13-s + (−1.22 + 1.57i)17-s + (0.637 − 0.770i)25-s + (−0.340 + 0.362i)29-s + (−1.57 + 0.934i)37-s + (−0.233 + 0.0922i)41-s + (−0.425 + 0.904i)45-s + (−0.951 + 0.309i)49-s + (1.91 + 0.557i)53-s + (−1.68 − 0.666i)61-s + (0.650 + 1.80i)65-s + (0.360 − 1.61i)73-s + (0.187 − 0.982i)81-s + ⋯
L(s)  = 1  + (0.904 − 0.425i)5-s + (−0.770 + 0.637i)9-s + (−0.180 + 1.91i)13-s + (−1.22 + 1.57i)17-s + (0.637 − 0.770i)25-s + (−0.340 + 0.362i)29-s + (−1.57 + 0.934i)37-s + (−0.233 + 0.0922i)41-s + (−0.425 + 0.904i)45-s + (−0.951 + 0.309i)49-s + (1.91 + 0.557i)53-s + (−1.68 − 0.666i)61-s + (0.650 + 1.80i)65-s + (0.360 − 1.61i)73-s + (0.187 − 0.982i)81-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=((0.0439−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0439 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s) L(s)=((0.0439−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0439 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: 0.0439−0.999i0.0439 - 0.999i
Analytic conductor: 1.996261.99626
Root analytic conductor: 1.412891.41289
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ4000(353,⋅)\chi_{4000} (353, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4000, ( :0), 0.0439−0.999i)(2,\ 4000,\ (\ :0),\ 0.0439 - 0.999i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1126085041.112608504
L(12)L(\frac12) ≈\approx 1.1126085041.112608504
L(1)L(1) 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+(−0.904+0.425i)T 1 + (-0.904 + 0.425i)T
good3 1+(0.770−0.637i)T2 1 + (0.770 - 0.637i)T^{2}
7 1+(0.951−0.309i)T2 1 + (0.951 - 0.309i)T^{2}
11 1+(−0.992−0.125i)T2 1 + (-0.992 - 0.125i)T^{2}
13 1+(0.180−1.91i)T+(−0.982−0.187i)T2 1 + (0.180 - 1.91i)T + (-0.982 - 0.187i)T^{2}
17 1+(1.22−1.57i)T+(−0.248−0.968i)T2 1 + (1.22 - 1.57i)T + (-0.248 - 0.968i)T^{2}
19 1+(0.637−0.770i)T2 1 + (0.637 - 0.770i)T^{2}
23 1+(−0.684+0.728i)T2 1 + (-0.684 + 0.728i)T^{2}
29 1+(0.340−0.362i)T+(−0.0627−0.998i)T2 1 + (0.340 - 0.362i)T + (-0.0627 - 0.998i)T^{2}
31 1+(0.968−0.248i)T2 1 + (0.968 - 0.248i)T^{2}
37 1+(1.57−0.934i)T+(0.481−0.876i)T2 1 + (1.57 - 0.934i)T + (0.481 - 0.876i)T^{2}
41 1+(0.233−0.0922i)T+(0.728−0.684i)T2 1 + (0.233 - 0.0922i)T + (0.728 - 0.684i)T^{2}
43 1+(−0.587−0.809i)T2 1 + (-0.587 - 0.809i)T^{2}
47 1+(0.368+0.929i)T2 1 + (0.368 + 0.929i)T^{2}
53 1+(−1.91−0.557i)T+(0.844+0.535i)T2 1 + (-1.91 - 0.557i)T + (0.844 + 0.535i)T^{2}
59 1+(0.425+0.904i)T2 1 + (0.425 + 0.904i)T^{2}
61 1+(1.68+0.666i)T+(0.728+0.684i)T2 1 + (1.68 + 0.666i)T + (0.728 + 0.684i)T^{2}
67 1+(−0.998−0.0627i)T2 1 + (-0.998 - 0.0627i)T^{2}
71 1+(−0.929+0.368i)T2 1 + (-0.929 + 0.368i)T^{2}
73 1+(−0.360+1.61i)T+(−0.904−0.425i)T2 1 + (-0.360 + 1.61i)T + (-0.904 - 0.425i)T^{2}
79 1+(0.637+0.770i)T2 1 + (0.637 + 0.770i)T^{2}
83 1+(−0.770−0.637i)T2 1 + (-0.770 - 0.637i)T^{2}
89 1+(−1.68+1.06i)T+(0.425−0.904i)T2 1 + (-1.68 + 1.06i)T + (0.425 - 0.904i)T^{2}
97 1+(−1.95+0.0613i)T+(0.998−0.0627i)T2 1 + (-1.95 + 0.0613i)T + (0.998 - 0.0627i)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

−8.874010955438326469863798047493, −8.337303618025485506285461060496, −7.25309960497273008311741067376, −6.39919206479778828177046974491, −6.03529669272984719113394541920, −4.93470315465420772828649275714, −4.49847630707836871438384014257, −3.38407848570269008447101557545, −2.02817979536375825319690222686, −1.81733992241110653377350178973, 0.56836580544248936518399511761, 2.19155316805167939535886148509, 2.87884832194065599491588422713, 3.58936438692619066186152882130, 4.98606595396468483822252876695, 5.46936325888064102077649216571, 6.18416975240049498803720118342, 6.96701814359599445873322334706, 7.60199337770892121189197065627, 8.670935245356631584137728087745

Graph of the ZZ-function along the critical line