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Properties

Label 2-4000-125.13-c0-0-0
Degree 22
Conductor 40004000
Sign 0.805−0.592i0.805 - 0.592i
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  + (0.998 + 0.0627i)5-s + (0.125 + 0.992i)9-s + (−1.45 − 1.12i)13-s + (0.115 + 1.22i)17-s + (0.992 + 0.125i)25-s + (1.65 + 1.05i)29-s + (0.313 + 0.461i)37-s + (1.35 + 0.742i)41-s + (0.0627 + 0.998i)45-s + (−0.587 − 0.809i)49-s + (0.400 + 0.173i)53-s + (1.74 − 0.961i)61-s + (−1.37 − 1.21i)65-s + (0.0540 + 1.72i)73-s + (−0.968 + 0.248i)81-s + ⋯
L(s)  = 1  + (0.998 + 0.0627i)5-s + (0.125 + 0.992i)9-s + (−1.45 − 1.12i)13-s + (0.115 + 1.22i)17-s + (0.992 + 0.125i)25-s + (1.65 + 1.05i)29-s + (0.313 + 0.461i)37-s + (1.35 + 0.742i)41-s + (0.0627 + 0.998i)45-s + (−0.587 − 0.809i)49-s + (0.400 + 0.173i)53-s + (1.74 − 0.961i)61-s + (−1.37 − 1.21i)65-s + (0.0540 + 1.72i)73-s + (−0.968 + 0.248i)81-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=((0.805−0.592i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s) L(s)=((0.805−0.592i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: 0.805−0.592i0.805 - 0.592i
Analytic conductor: 1.996261.99626
Root analytic conductor: 1.412891.41289
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ4000(513,⋅)\chi_{4000} (513, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4000, ( :0), 0.805−0.592i)(2,\ 4000,\ (\ :0),\ 0.805 - 0.592i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5108285961.510828596
L(12)L(\frac12) ≈\approx 1.5108285961.510828596
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+(−0.998−0.0627i)T 1 + (-0.998 - 0.0627i)T
good3 1+(−0.125−0.992i)T2 1 + (-0.125 - 0.992i)T^{2}
7 1+(0.587+0.809i)T2 1 + (0.587 + 0.809i)T^{2}
11 1+(−0.637−0.770i)T2 1 + (-0.637 - 0.770i)T^{2}
13 1+(1.45+1.12i)T+(0.248+0.968i)T2 1 + (1.45 + 1.12i)T + (0.248 + 0.968i)T^{2}
17 1+(−0.115−1.22i)T+(−0.982+0.187i)T2 1 + (-0.115 - 1.22i)T + (-0.982 + 0.187i)T^{2}
19 1+(0.992+0.125i)T2 1 + (0.992 + 0.125i)T^{2}
23 1+(0.844+0.535i)T2 1 + (0.844 + 0.535i)T^{2}
29 1+(−1.65−1.05i)T+(0.425+0.904i)T2 1 + (-1.65 - 1.05i)T + (0.425 + 0.904i)T^{2}
31 1+(−0.187−0.982i)T2 1 + (-0.187 - 0.982i)T^{2}
37 1+(−0.313−0.461i)T+(−0.368+0.929i)T2 1 + (-0.313 - 0.461i)T + (-0.368 + 0.929i)T^{2}
41 1+(−1.35−0.742i)T+(0.535+0.844i)T2 1 + (-1.35 - 0.742i)T + (0.535 + 0.844i)T^{2}
43 1+(0.951+0.309i)T2 1 + (0.951 + 0.309i)T^{2}
47 1+(0.481−0.876i)T2 1 + (0.481 - 0.876i)T^{2}
53 1+(−0.400−0.173i)T+(0.684+0.728i)T2 1 + (-0.400 - 0.173i)T + (0.684 + 0.728i)T^{2}
59 1+(−0.0627+0.998i)T2 1 + (-0.0627 + 0.998i)T^{2}
61 1+(−1.74+0.961i)T+(0.535−0.844i)T2 1 + (-1.74 + 0.961i)T + (0.535 - 0.844i)T^{2}
67 1+(0.904+0.425i)T2 1 + (0.904 + 0.425i)T^{2}
71 1+(0.876+0.481i)T2 1 + (0.876 + 0.481i)T^{2}
73 1+(−0.0540−1.72i)T+(−0.998+0.0627i)T2 1 + (-0.0540 - 1.72i)T + (-0.998 + 0.0627i)T^{2}
79 1+(0.992−0.125i)T2 1 + (0.992 - 0.125i)T^{2}
83 1+(0.125−0.992i)T2 1 + (0.125 - 0.992i)T^{2}
89 1+(1.23−1.31i)T+(−0.0627−0.998i)T2 1 + (1.23 - 1.31i)T + (-0.0627 - 0.998i)T^{2}
97 1+(0.436+1.95i)T+(−0.904+0.425i)T2 1 + (0.436 + 1.95i)T + (-0.904 + 0.425i)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

−8.469097518866360041246910705637, −8.135435194824706194904752879781, −7.19167664827508920665233782070, −6.52577756104116946032307709263, −5.54941526208267629478417721304, −5.15169915532339738420616685969, −4.31028928923584993768301284546, −2.95195749917509325243090907685, −2.40663841312632066953290398243, −1.33238240438493452323500456230, 0.923927910640083246920424549825, 2.27373952278604018903512389007, 2.79742748460050036153195778205, 4.13233933291251547941404418033, 4.78253412800832602722936549719, 5.60599373371858127891340378522, 6.46530090663555683690812610499, 6.93301279055503909094400413185, 7.69855555758460980729590990344, 8.812417774399156650407433007622

Graph of the ZZ-function along the critical line