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Properties

Label 2-2280-2280.1139-c0-0-8
Degree 22
Conductor 22802280
Sign ii
Analytic cond. 1.137861.13786
Root an. cond. 1.066701.06670
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  + (−0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s − 5-s + 1.00·6-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (0.707 − 0.707i)10-s − 2i·11-s + (−0.707 + 0.707i)12-s + 1.41i·13-s + (0.707 + 0.707i)15-s − 1.00·16-s + (−0.707 − 0.707i)18-s + 19-s + 1.00i·20-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.707 − 0.707i)3-s − 1.00i·4-s − 5-s + 1.00·6-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (0.707 − 0.707i)10-s − 2i·11-s + (−0.707 + 0.707i)12-s + 1.41i·13-s + (0.707 + 0.707i)15-s − 1.00·16-s + (−0.707 − 0.707i)18-s + 19-s + 1.00i·20-s + ⋯

Functional equation

Λ(s)=(2280s/2 ΓC(s) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2280s/2 ΓC(s) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22802280    =    23⋅3⋅5⋅192^{3} \cdot 3 \cdot 5 \cdot 19
Sign: ii
Analytic conductor: 1.137861.13786
Root analytic conductor: 1.066701.06670
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2280(1139,⋅)\chi_{2280} (1139, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2280, ( :0), i)(2,\ 2280,\ (\ :0),\ i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.37440148980.3744014898
L(12)L(\frac12) ≈\approx 0.37440148980.3744014898
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+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
3 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
5 1+T 1 + T
19 1−T 1 - T
good7 1+T2 1 + T^{2}
11 1+2iT−T2 1 + 2iT - T^{2}
13 1−1.41iT−T2 1 - 1.41iT - T^{2}
17 1+T2 1 + T^{2}
23 1−T2 1 - T^{2}
29 1−T2 1 - T^{2}
31 1+T2 1 + T^{2}
37 1+1.41iT−T2 1 + 1.41iT - T^{2}
41 1+T2 1 + T^{2}
43 1−T2 1 - T^{2}
47 1−T2 1 - T^{2}
53 1+1.41T+T2 1 + 1.41T + T^{2}
59 1+T2 1 + T^{2}
61 1+2iT−T2 1 + 2iT - T^{2}
67 1+1.41T+T2 1 + 1.41T + T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1+T2 1 + T^{2}
83 1+T2 1 + T^{2}
89 1+T2 1 + T^{2}
97 1−1.41T+T2 1 - 1.41T + 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

−8.782793213847560318708865958240, −8.103830672841535332111044268784, −7.51191659289841465985318996560, −6.73691923181224540774244912393, −6.11914392004775590811596700980, −5.33544874940124108914983072740, −4.41666077373413560886090769362, −3.17841979441556657397835481272, −1.61658708890865965416116467196, −0.42151937943283536197836364234, 1.21103871898110647874194347277, 2.84037024928082009626347137519, 3.60863320013442869746374920478, 4.57632090422184291193173700152, 5.04228661423271317844112600621, 6.45948293277654186146233724407, 7.41551759960314149018763014763, 7.77815250033549944614597397674, 8.813402087965620523207356172014, 9.636586037390363913669083266580

Graph of the ZZ-function along the critical line