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Properties

Label 2-2600-2600.2339-c0-0-1
Degree 22
Conductor 26002600
Sign 0.604+0.796i0.604 + 0.796i
Analytic cond. 1.297561.29756
Root an. cond. 1.139101.13910
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)2-s + (−1.64 − 0.535i)3-s + (−0.309 + 0.951i)4-s + (0.866 + 0.5i)5-s + (0.535 + 1.64i)6-s − 0.209i·7-s + (0.951 − 0.309i)8-s + (1.61 + 1.17i)9-s + (−0.104 − 0.994i)10-s + (1.01 − 1.40i)12-s + (0.587 − 0.809i)13-s + (−0.169 + 0.122i)14-s + (−1.15 − 1.28i)15-s + (−0.809 − 0.587i)16-s + (−0.395 + 0.128i)17-s − 2i·18-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−1.64 − 0.535i)3-s + (−0.309 + 0.951i)4-s + (0.866 + 0.5i)5-s + (0.535 + 1.64i)6-s − 0.209i·7-s + (0.951 − 0.309i)8-s + (1.61 + 1.17i)9-s + (−0.104 − 0.994i)10-s + (1.01 − 1.40i)12-s + (0.587 − 0.809i)13-s + (−0.169 + 0.122i)14-s + (−1.15 − 1.28i)15-s + (−0.809 − 0.587i)16-s + (−0.395 + 0.128i)17-s − 2i·18-s + ⋯

Functional equation

Λ(s)=(2600s/2 ΓC(s) L(s)=((0.604+0.796i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2600s/2 ΓC(s) L(s)=((0.604+0.796i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26002600    =    23⋅52⋅132^{3} \cdot 5^{2} \cdot 13
Sign: 0.604+0.796i0.604 + 0.796i
Analytic conductor: 1.297561.29756
Root analytic conductor: 1.139101.13910
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2600(2339,⋅)\chi_{2600} (2339, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2600, ( :0), 0.604+0.796i)(2,\ 2600,\ (\ :0),\ 0.604 + 0.796i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.58992336670.5899233667
L(12)L(\frac12) ≈\approx 0.58992336670.5899233667
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+(0.587+0.809i)T 1 + (0.587 + 0.809i)T
5 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
13 1+(−0.587+0.809i)T 1 + (-0.587 + 0.809i)T
good3 1+(1.64+0.535i)T+(0.809+0.587i)T2 1 + (1.64 + 0.535i)T + (0.809 + 0.587i)T^{2}
7 1+0.209iT−T2 1 + 0.209iT - T^{2}
11 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
17 1+(0.395−0.128i)T+(0.809−0.587i)T2 1 + (0.395 - 0.128i)T + (0.809 - 0.587i)T^{2}
19 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
23 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
29 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
31 1+(−0.363−1.11i)T+(−0.809+0.587i)T2 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2}
37 1+(0.786−1.08i)T+(−0.309−0.951i)T2 1 + (0.786 - 1.08i)T + (-0.309 - 0.951i)T^{2}
41 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
43 1+1.98iT−T2 1 + 1.98iT - T^{2}
47 1+(−1.86−0.604i)T+(0.809+0.587i)T2 1 + (-1.86 - 0.604i)T + (0.809 + 0.587i)T^{2}
53 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
59 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
61 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
67 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
71 1+(0.459−1.41i)T+(−0.809−0.587i)T2 1 + (0.459 - 1.41i)T + (-0.809 - 0.587i)T^{2}
73 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
79 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
83 1+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
89 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
97 1+(−0.809−0.587i)T2 1 + (-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

−9.087313023737302336435908754639, −8.260743596475637356470218898360, −7.13719263448814576632651267136, −6.83207562071336284802967475731, −5.81471274110815642914104932464, −5.25116526825653856332947938063, −4.17120260955079264268229189460, −2.98508018619587250605455662099, −1.84499775877002126331198670647, −0.917423013970033506730666986793, 0.847167621508938679712724457558, 2.02333808161014639497224636305, 4.16139376749144469295452461705, 4.72617059193881026843747394623, 5.58764712387040363408677630406, 6.01761206640922975151697449504, 6.59094002949309393372536005201, 7.44473654058901677270923060415, 8.654488724066997508098084523111, 9.237991370388709674717629274632

Graph of the ZZ-function along the critical line