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Properties

Label 2-2600-2600.2131-c0-0-3
Degree 22
Conductor 26002600
Sign 0.985−0.166i0.985 - 0.166i
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  + (0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.5 − 0.866i)5-s + (0.809 + 0.587i)6-s + 1.33·7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)10-s + (−0.309 + 0.951i)12-s + (0.309 − 0.951i)13-s + (0.413 + 1.27i)14-s + (−0.913 − 0.406i)15-s + (0.309 − 0.951i)16-s + (0.169 + 0.122i)17-s + (0.913 + 0.406i)20-s + (1.08 − 0.786i)21-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.5 − 0.866i)5-s + (0.809 + 0.587i)6-s + 1.33·7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)10-s + (−0.309 + 0.951i)12-s + (0.309 − 0.951i)13-s + (0.413 + 1.27i)14-s + (−0.913 − 0.406i)15-s + (0.309 − 0.951i)16-s + (0.169 + 0.122i)17-s + (0.913 + 0.406i)20-s + (1.08 − 0.786i)21-s + ⋯

Functional equation

Λ(s)=(2600s/2 ΓC(s) L(s)=((0.985−0.166i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2600s/2 ΓC(s) L(s)=((0.985−0.166i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26002600    =    23⋅52⋅132^{3} \cdot 5^{2} \cdot 13
Sign: 0.985−0.166i0.985 - 0.166i
Analytic conductor: 1.297561.29756
Root analytic conductor: 1.139101.13910
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2600(2131,⋅)\chi_{2600} (2131, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2600, ( :0), 0.985−0.166i)(2,\ 2600,\ (\ :0),\ 0.985 - 0.166i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7697833531.769783353
L(12)L(\frac12) ≈\approx 1.7697833531.769783353
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+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
13 1+(−0.309+0.951i)T 1 + (-0.309 + 0.951i)T
good3 1+(−0.809+0.587i)T+(0.309−0.951i)T2 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2}
7 1−1.33T+T2 1 - 1.33T + T^{2}
11 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
17 1+(−0.169−0.122i)T+(0.309+0.951i)T2 1 + (-0.169 - 0.122i)T + (0.309 + 0.951i)T^{2}
19 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
23 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
29 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
31 1+(0.5+0.363i)T+(0.309+0.951i)T2 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2}
37 1+(−0.564+1.73i)T+(−0.809−0.587i)T2 1 + (-0.564 + 1.73i)T + (-0.809 - 0.587i)T^{2}
41 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
43 1−1.33T+T2 1 - 1.33T + T^{2}
47 1+(−0.169+0.122i)T+(0.309−0.951i)T2 1 + (-0.169 + 0.122i)T + (0.309 - 0.951i)T^{2}
53 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
59 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
61 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
67 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
71 1+(1.47−1.07i)T+(0.309−0.951i)T2 1 + (1.47 - 1.07i)T + (0.309 - 0.951i)T^{2}
73 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
79 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
83 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
89 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
97 1+(−0.309+0.951i)T2 1 + (-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

−8.794654005765510197265018713533, −7.965812402314156859802200305920, −7.84514078429791429197004184712, −7.17527651105145007364994076130, −5.78392878068079831393948097871, −5.29824002310342934062048647594, −4.42325859647228440268252745254, −3.66412881170457221842255374332, −2.43419702547195889666490182996, −1.13281841170891587346248486963, 1.51330011689573335228657235293, 2.56498409482361125958096431462, 3.30667326977277782127220017602, 4.20595576689375830416824951184, 4.58648474974934061051980905485, 5.76594941231272729031560532152, 6.75145299354972870915829144772, 7.84393207410874481443298337243, 8.425037727565245227167278164438, 9.142420846205221024572407983303

Graph of the ZZ-function along the critical line