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Properties

Label 2-2600-2600.1091-c0-0-3
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  + (0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.5 + 0.866i)5-s + (0.309 − 0.951i)6-s + 0.209·7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)10-s + (0.809 − 0.587i)12-s + (0.809 − 0.587i)13-s + (0.169 + 0.122i)14-s + (0.669 − 0.743i)15-s + (−0.809 + 0.587i)16-s + (−0.604 + 1.86i)17-s + (−0.669 + 0.743i)20-s + (−0.0646 − 0.198i)21-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 + 0.951i)4-s + (0.5 + 0.866i)5-s + (0.309 − 0.951i)6-s + 0.209·7-s + (−0.309 + 0.951i)8-s + (−0.104 + 0.994i)10-s + (0.809 − 0.587i)12-s + (0.809 − 0.587i)13-s + (0.169 + 0.122i)14-s + (0.669 − 0.743i)15-s + (−0.809 + 0.587i)16-s + (−0.604 + 1.86i)17-s + (−0.669 + 0.743i)20-s + (−0.0646 − 0.198i)21-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(1091,⋅)\chi_{2600} (1091, \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 1.9845452511.984545251
L(12)L(\frac12) ≈\approx 1.9845452511.984545251
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+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
5 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
13 1+(−0.809+0.587i)T 1 + (-0.809 + 0.587i)T
good3 1+(0.309+0.951i)T+(−0.809+0.587i)T2 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2}
7 1−0.209T+T2 1 - 0.209T + T^{2}
11 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
17 1+(0.604−1.86i)T+(−0.809−0.587i)T2 1 + (0.604 - 1.86i)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+(−0.5+1.53i)T+(−0.809−0.587i)T2 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2}
37 1+(−1.08+0.786i)T+(0.309−0.951i)T2 1 + (-1.08 + 0.786i)T + (0.309 - 0.951i)T^{2}
41 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
43 1+0.209T+T2 1 + 0.209T + T^{2}
47 1+(−0.604−1.86i)T+(−0.809+0.587i)T2 1 + (-0.604 - 1.86i)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+(0.413+1.27i)T+(−0.809+0.587i)T2 1 + (0.413 + 1.27i)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

−8.929552080209667701559174666628, −7.910457969348046967544524559556, −7.64112378812113378471144616109, −6.57254419971491815371155506973, −6.12157977110801661170824127245, −5.80087172539769623501617847675, −4.39058661911185907421266775820, −3.64421445832829662485371011364, −2.53899628218013657918421247473, −1.63052232284242745040285389947, 1.17700883407338429930810536752, 2.29781808674198310552429290001, 3.44810014062428063497923132788, 4.44318092725068435045112300417, 4.83796012087468033544371435806, 5.43768513778952520158699161131, 6.38184806224669868836192324349, 7.17975173253572534292645553431, 8.565995313950494648205671754897, 9.174977091239441954253273558647

Graph of the ZZ-function along the critical line