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Properties

Label 2-882-1.1-c5-0-25
Degree 22
Conductor 882882
Sign 11
Analytic cond. 141.458141.458
Root an. cond. 11.893611.8936
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 16·4-s + 6·5-s − 64·8-s − 24·10-s + 666·11-s − 559·13-s + 256·16-s + 1.74e3·17-s + 1.15e3·19-s + 96·20-s − 2.66e3·22-s + 3.46e3·23-s − 3.08e3·25-s + 2.23e3·26-s − 3.37e3·29-s + 6.29e3·31-s − 1.02e3·32-s − 6.96e3·34-s + 3.13e3·37-s − 4.62e3·38-s − 384·40-s + 4.86e3·41-s − 1.14e4·43-s + 1.06e4·44-s − 1.38e4·46-s − 2.31e3·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.107·5-s − 0.353·8-s − 0.0758·10-s + 1.65·11-s − 0.917·13-s + 1/4·16-s + 1.46·17-s + 0.735·19-s + 0.0536·20-s − 1.17·22-s + 1.36·23-s − 0.988·25-s + 0.648·26-s − 0.744·29-s + 1.17·31-s − 0.176·32-s − 1.03·34-s + 0.375·37-s − 0.519·38-s − 0.0379·40-s + 0.452·41-s − 0.940·43-s + 0.829·44-s − 0.966·46-s − 0.152·47-s + ⋯

Functional equation

Λ(s)=(882s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(882s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 882882    =    2⋅32⋅722 \cdot 3^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 141.458141.458
Root analytic conductor: 11.893611.8936
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 882, ( :5/2), 1)(2,\ 882,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 2.0037417232.003741723
L(12)L(\frac12) ≈\approx 2.0037417232.003741723
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 1
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

−9.664349138323203778659196575783, −8.658495115017593443378072223597, −7.66557689884245196807138029878, −7.02076879289246138666925747518, −6.08333780203388826245278855734, −5.11487495718468250589765372955, −3.85265685426776837937786103072, −2.87139104820102573209725856815, −1.55514932837920553532004312843, −0.76314077607542482855287364593, 0.76314077607542482855287364593, 1.55514932837920553532004312843, 2.87139104820102573209725856815, 3.85265685426776837937786103072, 5.11487495718468250589765372955, 6.08333780203388826245278855734, 7.02076879289246138666925747518, 7.66557689884245196807138029878, 8.658495115017593443378072223597, 9.664349138323203778659196575783

Graph of the ZZ-function along the critical line