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Properties

Label 2-294-1.1-c5-0-27
Degree 22
Conductor 294294
Sign −1-1
Analytic cond. 47.152847.1528
Root an. cond. 6.866796.86679
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s − 9·3-s + 16·4-s + 6·5-s − 36·6-s + 64·8-s + 81·9-s + 24·10-s − 666·11-s − 144·12-s + 559·13-s − 54·15-s + 256·16-s + 1.74e3·17-s + 324·18-s − 1.15e3·19-s + 96·20-s − 2.66e3·22-s − 3.46e3·23-s − 576·24-s − 3.08e3·25-s + 2.23e3·26-s − 729·27-s + 3.37e3·29-s − 216·30-s − 6.29e3·31-s + 1.02e3·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.107·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.0758·10-s − 1.65·11-s − 0.288·12-s + 0.917·13-s − 0.0619·15-s + 1/4·16-s + 1.46·17-s + 0.235·18-s − 0.735·19-s + 0.0536·20-s − 1.17·22-s − 1.36·23-s − 0.204·24-s − 0.988·25-s + 0.648·26-s − 0.192·27-s + 0.744·29-s − 0.0438·30-s − 1.17·31-s + 0.176·32-s + ⋯

Functional equation

Λ(s)=(294s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(294s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    2⋅3⋅722 \cdot 3 \cdot 7^{2}
Sign: −1-1
Analytic conductor: 47.152847.1528
Root analytic conductor: 6.866796.86679
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 294, ( :5/2), −1)(2,\ 294,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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−p2T 1 - p^{2} T
3 1+p2T 1 + p^{2} T
7 1 1
good5 1−6T+p5T2 1 - 6 T + p^{5} T^{2}
11 1+666T+p5T2 1 + 666 T + p^{5} T^{2}
13 1−43pT+p5T2 1 - 43 p T + p^{5} T^{2}
17 1−1740T+p5T2 1 - 1740 T + p^{5} T^{2}
19 1+1157T+p5T2 1 + 1157 T + p^{5} T^{2}
23 1+3468T+p5T2 1 + 3468 T + p^{5} T^{2}
29 1−3372T+p5T2 1 - 3372 T + p^{5} T^{2}
31 1+203pT+p5T2 1 + 203 p T + p^{5} T^{2}
37 1−3131T+p5T2 1 - 3131 T + p^{5} T^{2}
41 1−4866T+p5T2 1 - 4866 T + p^{5} T^{2}
43 1+11407T+p5T2 1 + 11407 T + p^{5} T^{2}
47 1+2310T+p5T2 1 + 2310 T + p^{5} T^{2}
53 1+28296T+p5T2 1 + 28296 T + p^{5} T^{2}
59 1+20544T+p5T2 1 + 20544 T + p^{5} T^{2}
61 1−4630T+p5T2 1 - 4630 T + p^{5} T^{2}
67 1+18745T+p5T2 1 + 18745 T + p^{5} T^{2}
71 1+38226T+p5T2 1 + 38226 T + p^{5} T^{2}
73 1+70589T+p5T2 1 + 70589 T + p^{5} T^{2}
79 1+62293T+p5T2 1 + 62293 T + p^{5} T^{2}
83 1+79818T+p5T2 1 + 79818 T + p^{5} T^{2}
89 1−18120T+p5T2 1 - 18120 T + p^{5} T^{2}
97 1+124754T+p5T2 1 + 124754 T + p^{5} 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

−10.54993920027823863704108978803, −9.931304251034525736123619334239, −8.254027796626366659307480027620, −7.50177768939488291446826229838, −6.06663401348382362301575337234, −5.57443980472509031880262589118, −4.37498243713819438479013534453, −3.13832296735515526073587324644, −1.70962022664517275485913801710, 0, 1.70962022664517275485913801710, 3.13832296735515526073587324644, 4.37498243713819438479013534453, 5.57443980472509031880262589118, 6.06663401348382362301575337234, 7.50177768939488291446826229838, 8.254027796626366659307480027620, 9.931304251034525736123619334239, 10.54993920027823863704108978803

Graph of the ZZ-function along the critical line