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Properties

Label 2-4000-125.22-c0-0-0
Degree 22
Conductor 40004000
Sign 0.999+0.0439i0.999 + 0.0439i
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 + (0.555 + 0.0525i)13-s + (0.148 + 0.115i)17-s + (0.637 − 0.770i)25-s + (−0.340 + 0.362i)29-s + (−0.404 − 0.683i)37-s + (0.233 − 0.0922i)41-s + (−0.425 + 0.904i)45-s + (0.951 − 0.309i)49-s + (0.0175 − 0.0603i)53-s + (1.68 + 0.666i)61-s + (−0.524 + 0.189i)65-s + (1.09 + 0.245i)73-s + (0.187 − 0.982i)81-s + ⋯
L(s)  = 1  + (−0.904 + 0.425i)5-s + (0.770 − 0.637i)9-s + (0.555 + 0.0525i)13-s + (0.148 + 0.115i)17-s + (0.637 − 0.770i)25-s + (−0.340 + 0.362i)29-s + (−0.404 − 0.683i)37-s + (0.233 − 0.0922i)41-s + (−0.425 + 0.904i)45-s + (0.951 − 0.309i)49-s + (0.0175 − 0.0603i)53-s + (1.68 + 0.666i)61-s + (−0.524 + 0.189i)65-s + (1.09 + 0.245i)73-s + (0.187 − 0.982i)81-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: 0.999+0.0439i0.999 + 0.0439i
Analytic conductor: 1.996261.99626
Root analytic conductor: 1.412891.41289
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ4000(897,⋅)\chi_{4000} (897, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4000, ( :0), 0.999+0.0439i)(2,\ 4000,\ (\ :0),\ 0.999 + 0.0439i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1876062481.187606248
L(12)L(\frac12) ≈\approx 1.1876062481.187606248
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+(−0.555−0.0525i)T+(0.982+0.187i)T2 1 + (-0.555 - 0.0525i)T + (0.982 + 0.187i)T^{2}
17 1+(−0.148−0.115i)T+(0.248+0.968i)T2 1 + (-0.148 - 0.115i)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+(0.404+0.683i)T+(−0.481+0.876i)T2 1 + (0.404 + 0.683i)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+(−0.0175+0.0603i)T+(−0.844−0.535i)T2 1 + (-0.0175 + 0.0603i)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+(−1.09−0.245i)T+(0.904+0.425i)T2 1 + (-1.09 - 0.245i)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+(0.0137+0.436i)T+(−0.998+0.0627i)T2 1 + (0.0137 + 0.436i)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

−8.635657497459925266579849385477, −7.78738895912170723968354377010, −7.16338554340347677225887969595, −6.58944944740045599805596410142, −5.73593379393305119391555410804, −4.71998713997281095182706370104, −3.84406610784438934985087056451, −3.46875807111809548420072972950, −2.23034026282355052912704346301, −0.915823152696376132682575643525, 0.999606620217287016226190991988, 2.14038891335363813846497348116, 3.37233949070061132517144875034, 4.07761999054164066234168643214, 4.80429926087061022184956794349, 5.52244082741293673388484431838, 6.56820303495944578365519835708, 7.29089800720330840724243054872, 7.931636386587508908325756121990, 8.475801249500912044976708730325

Graph of the ZZ-function along the critical line