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Properties

Label 2-40e2-1.1-c3-0-15
Degree 22
Conductor 16001600
Sign 11
Analytic cond. 94.403094.4030
Root an. cond. 9.716129.71612
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27·9-s − 92·13-s − 104·17-s − 130·29-s + 396·37-s + 230·41-s − 343·49-s + 572·53-s + 830·61-s − 592·73-s + 729·81-s + 1.67e3·89-s − 1.81e3·97-s − 598·101-s + 1.74e3·109-s − 1.32e3·113-s + 2.48e3·117-s + ⋯
L(s)  = 1  − 9-s − 1.96·13-s − 1.48·17-s − 0.832·29-s + 1.75·37-s + 0.876·41-s − 49-s + 1.48·53-s + 1.74·61-s − 0.949·73-s + 81-s + 1.98·89-s − 1.90·97-s − 0.589·101-s + 1.53·109-s − 1.10·113-s + 1.96·117-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 11
Analytic conductor: 94.403094.4030
Root analytic conductor: 9.716129.71612
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1600, ( :3/2), 1)(2,\ 1600,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.97078771970.9707877197
L(12)L(\frac12) ≈\approx 0.97078771970.9707877197
L(52)L(\frac{5}{2}) 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 1
good3 1+p3T2 1 + p^{3} T^{2}
7 1+p3T2 1 + p^{3} T^{2}
11 1+p3T2 1 + p^{3} T^{2}
13 1+92T+p3T2 1 + 92 T + p^{3} T^{2}
17 1+104T+p3T2 1 + 104 T + p^{3} T^{2}
19 1+p3T2 1 + p^{3} T^{2}
23 1+p3T2 1 + p^{3} T^{2}
29 1+130T+p3T2 1 + 130 T + p^{3} T^{2}
31 1+p3T2 1 + p^{3} T^{2}
37 1−396T+p3T2 1 - 396 T + p^{3} T^{2}
41 1−230T+p3T2 1 - 230 T + p^{3} T^{2}
43 1+p3T2 1 + p^{3} T^{2}
47 1+p3T2 1 + p^{3} T^{2}
53 1−572T+p3T2 1 - 572 T + p^{3} T^{2}
59 1+p3T2 1 + p^{3} T^{2}
61 1−830T+p3T2 1 - 830 T + p^{3} T^{2}
67 1+p3T2 1 + p^{3} T^{2}
71 1+p3T2 1 + p^{3} T^{2}
73 1+592T+p3T2 1 + 592 T + p^{3} T^{2}
79 1+p3T2 1 + p^{3} T^{2}
83 1+p3T2 1 + p^{3} T^{2}
89 1−1670T+p3T2 1 - 1670 T + p^{3} T^{2}
97 1+1816T+p3T2 1 + 1816 T + p^{3} 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

−9.153414178536364170474972581429, −8.236887585124414802485640997880, −7.45721832291116743968145268197, −6.67893136648116745588846640316, −5.73172590299788706605911013151, −4.92189185495959417070482783843, −4.09058620606015797332101007998, −2.73650619566920242331990061996, −2.20328927761097337870816619045, −0.44360603596238490082235246168, 0.44360603596238490082235246168, 2.20328927761097337870816619045, 2.73650619566920242331990061996, 4.09058620606015797332101007998, 4.92189185495959417070482783843, 5.73172590299788706605911013151, 6.67893136648116745588846640316, 7.45721832291116743968145268197, 8.236887585124414802485640997880, 9.153414178536364170474972581429

Graph of the ZZ-function along the critical line