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Properties

Label 2-40e2-100.31-c0-0-0
Degree 22
Conductor 16001600
Sign −0.368−0.929i-0.368 - 0.929i
Analytic cond. 0.7985040.798504
Root an. cond. 0.8935900.893590
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  + (0.587 + 0.809i)3-s + (−0.309 + 0.951i)5-s + 0.618i·7-s + (−0.190 + 0.587i)13-s + (−0.951 + 0.309i)15-s + (−0.951 + 1.30i)19-s + (−0.500 + 0.363i)21-s + (−0.951 + 0.309i)23-s + (−0.809 − 0.587i)25-s + (0.951 − 0.309i)27-s + (1.30 − 0.951i)29-s + (0.587 − 0.809i)31-s + (−0.587 − 0.190i)35-s + (0.309 − 0.951i)37-s + (−0.587 + 0.190i)39-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)3-s + (−0.309 + 0.951i)5-s + 0.618i·7-s + (−0.190 + 0.587i)13-s + (−0.951 + 0.309i)15-s + (−0.951 + 1.30i)19-s + (−0.500 + 0.363i)21-s + (−0.951 + 0.309i)23-s + (−0.809 − 0.587i)25-s + (0.951 − 0.309i)27-s + (1.30 − 0.951i)29-s + (0.587 − 0.809i)31-s + (−0.587 − 0.190i)35-s + (0.309 − 0.951i)37-s + (−0.587 + 0.190i)39-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((−0.368−0.929i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s) L(s)=((−0.368−0.929i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: −0.368−0.929i-0.368 - 0.929i
Analytic conductor: 0.7985040.798504
Root analytic conductor: 0.8935900.893590
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1600(831,⋅)\chi_{1600} (831, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1600, ( :0), −0.368−0.929i)(2,\ 1600,\ (\ :0),\ -0.368 - 0.929i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2035476621.203547662
L(12)L(\frac12) ≈\approx 1.2035476621.203547662
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+(0.309−0.951i)T 1 + (0.309 - 0.951i)T
good3 1+(−0.587−0.809i)T+(−0.309+0.951i)T2 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2}
7 1−0.618iT−T2 1 - 0.618iT - T^{2}
11 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
13 1+(0.190−0.587i)T+(−0.809−0.587i)T2 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2}
17 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
19 1+(0.951−1.30i)T+(−0.309−0.951i)T2 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2}
23 1+(0.951−0.309i)T+(0.809−0.587i)T2 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2}
29 1+(−1.30+0.951i)T+(0.309−0.951i)T2 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2}
31 1+(−0.587+0.809i)T+(−0.309−0.951i)T2 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2}
37 1+(−0.309+0.951i)T+(−0.809−0.587i)T2 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2}
41 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
43 1−1.61iT−T2 1 - 1.61iT - T^{2}
47 1+(−0.363−0.5i)T+(−0.309+0.951i)T2 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2}
53 1+(0.809−0.587i)T+(0.309−0.951i)T2 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2}
59 1+(1.53+0.5i)T+(0.809+0.587i)T2 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2}
61 1+(0.309+0.951i)T+(−0.809+0.587i)T2 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2}
67 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
71 1+(0.363+0.5i)T+(−0.309+0.951i)T2 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2}
73 1+(−0.309−0.951i)T+(−0.809+0.587i)T2 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2}
79 1+(−0.951−1.30i)T+(−0.309+0.951i)T2 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2}
83 1+(−0.587+0.809i)T+(−0.309−0.951i)T2 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2}
89 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
97 1+(−0.5+0.363i)T+(0.309−0.951i)T2 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)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

−9.809001227361923247004439598759, −9.224272471307720745956932795526, −8.194567102398282497923174291592, −7.73924018735133681392093578133, −6.35551578589057793637075630985, −6.09784496662361021465195119874, −4.50667966400483159496300568566, −3.97879710914419698497383099839, −2.99392619603187657659906976570, −2.12933265333839861037575657740, 0.906349613690973228259059539696, 2.11868694300867345995657015303, 3.21307347883301428800597200976, 4.46675729343005338010062467282, 4.99702864640554786271150938351, 6.32660077880599792239766290920, 7.12208267590452753699829002608, 7.82864318467304611372112415787, 8.534897087310013864931274552520, 8.975510094366421031631099485596

Graph of the ZZ-function along the critical line