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Properties

Label 2-4000-125.2-c0-0-0
Degree 22
Conductor 40004000
Sign −0.303−0.952i-0.303 - 0.952i
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  + (−0.844 − 0.535i)5-s + (−0.904 − 0.425i)9-s + (−0.344 − 0.957i)13-s + (−1.24 + 1.41i)17-s + (0.425 + 0.904i)25-s + (−0.0922 + 0.233i)29-s + (1.01 + 1.30i)37-s + (0.374 + 1.96i)41-s + (0.535 + 0.844i)45-s + (−0.587 + 0.809i)49-s + (−0.404 + 0.683i)53-s + (0.316 − 1.65i)61-s + (−0.222 + 0.993i)65-s + (−1.91 + 0.555i)73-s + (0.637 + 0.770i)81-s + ⋯
L(s)  = 1  + (−0.844 − 0.535i)5-s + (−0.904 − 0.425i)9-s + (−0.344 − 0.957i)13-s + (−1.24 + 1.41i)17-s + (0.425 + 0.904i)25-s + (−0.0922 + 0.233i)29-s + (1.01 + 1.30i)37-s + (0.374 + 1.96i)41-s + (0.535 + 0.844i)45-s + (−0.587 + 0.809i)49-s + (−0.404 + 0.683i)53-s + (0.316 − 1.65i)61-s + (−0.222 + 0.993i)65-s + (−1.91 + 0.555i)73-s + (0.637 + 0.770i)81-s + ⋯

Functional equation

Λ(s)=(4000s/2 ΓC(s) L(s)=((−0.303−0.952i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(4000s/2 ΓC(s) L(s)=((−0.303−0.952i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40004000    =    25⋅532^{5} \cdot 5^{3}
Sign: −0.303−0.952i-0.303 - 0.952i
Analytic conductor: 1.996261.99626
Root analytic conductor: 1.412891.41289
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ4000(1377,⋅)\chi_{4000} (1377, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 4000, ( :0), −0.303−0.952i)(2,\ 4000,\ (\ :0),\ -0.303 - 0.952i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.37517446760.3751744676
L(12)L(\frac12) ≈\approx 0.37517446760.3751744676
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+(0.844+0.535i)T 1 + (0.844 + 0.535i)T
good3 1+(0.904+0.425i)T2 1 + (0.904 + 0.425i)T^{2}
7 1+(0.587−0.809i)T2 1 + (0.587 - 0.809i)T^{2}
11 1+(0.0627−0.998i)T2 1 + (0.0627 - 0.998i)T^{2}
13 1+(0.344+0.957i)T+(−0.770+0.637i)T2 1 + (0.344 + 0.957i)T + (-0.770 + 0.637i)T^{2}
17 1+(1.24−1.41i)T+(−0.125−0.992i)T2 1 + (1.24 - 1.41i)T + (-0.125 - 0.992i)T^{2}
19 1+(0.425+0.904i)T2 1 + (0.425 + 0.904i)T^{2}
23 1+(−0.368+0.929i)T2 1 + (-0.368 + 0.929i)T^{2}
29 1+(0.0922−0.233i)T+(−0.728−0.684i)T2 1 + (0.0922 - 0.233i)T + (-0.728 - 0.684i)T^{2}
31 1+(−0.992+0.125i)T2 1 + (-0.992 + 0.125i)T^{2}
37 1+(−1.01−1.30i)T+(−0.248+0.968i)T2 1 + (-1.01 - 1.30i)T + (-0.248 + 0.968i)T^{2}
41 1+(−0.374−1.96i)T+(−0.929+0.368i)T2 1 + (-0.374 - 1.96i)T + (-0.929 + 0.368i)T^{2}
43 1+(0.951−0.309i)T2 1 + (0.951 - 0.309i)T^{2}
47 1+(0.982−0.187i)T2 1 + (0.982 - 0.187i)T^{2}
53 1+(0.404−0.683i)T+(−0.481−0.876i)T2 1 + (0.404 - 0.683i)T + (-0.481 - 0.876i)T^{2}
59 1+(−0.535+0.844i)T2 1 + (-0.535 + 0.844i)T^{2}
61 1+(−0.316+1.65i)T+(−0.929−0.368i)T2 1 + (-0.316 + 1.65i)T + (-0.929 - 0.368i)T^{2}
67 1+(0.684+0.728i)T2 1 + (0.684 + 0.728i)T^{2}
71 1+(−0.187−0.982i)T2 1 + (-0.187 - 0.982i)T^{2}
73 1+(1.91−0.555i)T+(0.844−0.535i)T2 1 + (1.91 - 0.555i)T + (0.844 - 0.535i)T^{2}
79 1+(0.425−0.904i)T2 1 + (0.425 - 0.904i)T^{2}
83 1+(−0.904+0.425i)T2 1 + (-0.904 + 0.425i)T^{2}
89 1+(−0.659+1.19i)T+(−0.535−0.844i)T2 1 + (-0.659 + 1.19i)T + (-0.535 - 0.844i)T^{2}
97 1+(0.221+0.512i)T+(−0.684+0.728i)T2 1 + (0.221 + 0.512i)T + (-0.684 + 0.728i)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

−8.684804917163130566886089935282, −8.142735320545064198340885822909, −7.65334394031779652968438259773, −6.47720940698527892601753440096, −6.02273111608300230098562413926, −4.97419578268359161652012895554, −4.36264361376548981914051923053, −3.42454486767874358746988946629, −2.69099249720673125715962066697, −1.26552875340877029170429066174, 0.21500873562656545217412207866, 2.21034426263210305262649357584, 2.75588940027289519879293057650, 3.87342462417222137202887386289, 4.53837532208357406626792481300, 5.37528767701219904563464732098, 6.31312190435701965654460918397, 7.14150602706473231239388078450, 7.46992177007302234214101629501, 8.495791392465034416927620109737

Graph of the ZZ-function along the critical line