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Properties

Label 2-40e2-100.91-c0-0-0
Degree 22
Conductor 16001600
Sign −0.481−0.876i-0.481 - 0.876i
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  + (−0.951 + 0.309i)3-s + (0.809 − 0.587i)5-s + 1.61i·7-s + (−1.30 + 0.951i)13-s + (−0.587 + 0.809i)15-s + (−0.587 − 0.190i)19-s + (−0.500 − 1.53i)21-s + (−0.587 + 0.809i)23-s + (0.309 − 0.951i)25-s + (0.587 − 0.809i)27-s + (0.190 + 0.587i)29-s + (−0.951 − 0.309i)31-s + (0.951 + 1.30i)35-s + (−0.809 + 0.587i)37-s + (0.951 − 1.30i)39-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)3-s + (0.809 − 0.587i)5-s + 1.61i·7-s + (−1.30 + 0.951i)13-s + (−0.587 + 0.809i)15-s + (−0.587 − 0.190i)19-s + (−0.500 − 1.53i)21-s + (−0.587 + 0.809i)23-s + (0.309 − 0.951i)25-s + (0.587 − 0.809i)27-s + (0.190 + 0.587i)29-s + (−0.951 − 0.309i)31-s + (0.951 + 1.30i)35-s + (−0.809 + 0.587i)37-s + (0.951 − 1.30i)39-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((−0.481−0.876i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s) L(s)=((−0.481−0.876i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: −0.481−0.876i-0.481 - 0.876i
Analytic conductor: 0.7985040.798504
Root analytic conductor: 0.8935900.893590
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1600(191,⋅)\chi_{1600} (191, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1600, ( :0), −0.481−0.876i)(2,\ 1600,\ (\ :0),\ -0.481 - 0.876i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.62921676200.6292167620
L(12)L(\frac12) ≈\approx 0.62921676200.6292167620
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+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
good3 1+(0.951−0.309i)T+(0.809−0.587i)T2 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2}
7 1−1.61iT−T2 1 - 1.61iT - T^{2}
11 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
13 1+(1.30−0.951i)T+(0.309−0.951i)T2 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2}
17 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
19 1+(0.587+0.190i)T+(0.809+0.587i)T2 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2}
23 1+(0.587−0.809i)T+(−0.309−0.951i)T2 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2}
29 1+(−0.190−0.587i)T+(−0.809+0.587i)T2 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2}
31 1+(0.951+0.309i)T+(0.809+0.587i)T2 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2}
37 1+(0.809−0.587i)T+(0.309−0.951i)T2 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2}
41 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
43 1−0.618iT−T2 1 - 0.618iT - T^{2}
47 1+(−1.53+0.5i)T+(0.809−0.587i)T2 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2}
53 1+(−0.309−0.951i)T+(−0.809+0.587i)T2 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2}
59 1+(−0.363−0.5i)T+(−0.309+0.951i)T2 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2}
61 1+(−0.809−0.587i)T+(0.309+0.951i)T2 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2}
67 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
71 1+(1.53−0.5i)T+(0.809−0.587i)T2 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2}
73 1+(0.809+0.587i)T+(0.309+0.951i)T2 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2}
79 1+(−0.587+0.190i)T+(0.809−0.587i)T2 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2}
83 1+(0.951+0.309i)T+(0.809+0.587i)T2 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2}
89 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
97 1+(−0.5−1.53i)T+(−0.809+0.587i)T2 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)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

−9.838088196271625464117919703082, −9.025815959157027317747516900662, −8.669052122839764660360191335202, −7.35898576839707447428980619559, −6.29977771566192042382548132901, −5.65520188947811627115000272402, −5.16146807250773369423992958735, −4.37787100980143940373335555199, −2.63601913108886195616824733752, −1.88760119127643573341215305044, 0.52204646233237342968962369278, 2.07436226681351332262030798644, 3.30988391685286884074583068171, 4.42592180628927913427954571332, 5.41737684590961161715778816135, 6.08015600499275755730860184357, 7.08423195379189921771599361265, 7.26163154048967855301474884764, 8.470376387066196046239116137976, 9.676063369975670327358227614116

Graph of the ZZ-function along the critical line