↑

Properties

Label 2-882-1.1-c5-0-24
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 16·4-s + 26·5-s + 64·8-s + 104·10-s − 664·11-s − 318·13-s + 256·16-s + 1.58e3·17-s − 236·19-s + 416·20-s − 2.65e3·22-s − 2.21e3·23-s − 2.44e3·25-s − 1.27e3·26-s + 4.95e3·29-s + 7.12e3·31-s + 1.02e3·32-s + 6.32e3·34-s + 4.35e3·37-s − 944·38-s + 1.66e3·40-s + 1.05e4·41-s − 8.45e3·43-s − 1.06e4·44-s − 8.84e3·46-s + 5.35e3·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.465·5-s + 0.353·8-s + 0.328·10-s − 1.65·11-s − 0.521·13-s + 1/4·16-s + 1.32·17-s − 0.149·19-s + 0.232·20-s − 1.16·22-s − 0.871·23-s − 0.783·25-s − 0.369·26-s + 1.09·29-s + 1.33·31-s + 0.176·32-s + 0.938·34-s + 0.523·37-s − 0.106·38-s + 0.164·40-s + 0.979·41-s − 0.697·43-s − 0.827·44-s − 0.616·46-s + 0.353·47-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 3.5624531573.562453157
L(12)L(\frac12) ≈\approx 3.5624531573.562453157
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 1−26T+p5T2 1 - 26 T + p^{5} T^{2}
11 1+664T+p5T2 1 + 664 T + p^{5} T^{2}
13 1+318T+p5T2 1 + 318 T + p^{5} T^{2}
17 1−1582T+p5T2 1 - 1582 T + p^{5} T^{2}
19 1+236T+p5T2 1 + 236 T + p^{5} T^{2}
23 1+2212T+p5T2 1 + 2212 T + p^{5} T^{2}
29 1−4954T+p5T2 1 - 4954 T + p^{5} T^{2}
31 1−7128T+p5T2 1 - 7128 T + p^{5} T^{2}
37 1−4358T+p5T2 1 - 4358 T + p^{5} T^{2}
41 1−10542T+p5T2 1 - 10542 T + p^{5} T^{2}
43 1+8452T+p5T2 1 + 8452 T + p^{5} T^{2}
47 1−5352T+p5T2 1 - 5352 T + p^{5} T^{2}
53 1−33354T+p5T2 1 - 33354 T + p^{5} T^{2}
59 1+15436T+p5T2 1 + 15436 T + p^{5} T^{2}
61 1−36762T+p5T2 1 - 36762 T + p^{5} T^{2}
67 1−40972T+p5T2 1 - 40972 T + p^{5} T^{2}
71 1−9092T+p5T2 1 - 9092 T + p^{5} T^{2}
73 1−73454T+p5T2 1 - 73454 T + p^{5} T^{2}
79 1−89400T+p5T2 1 - 89400 T + p^{5} T^{2}
83 1+6428T+p5T2 1 + 6428 T + p^{5} T^{2}
89 1+122658T+p5T2 1 + 122658 T + p^{5} T^{2}
97 1+21370T+p5T2 1 + 21370 T + p^{5} T^{2}
show more
show less
   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

−9.830204412266161579588226374633, −8.200904634741764565590055787780, −7.79519147296136066151975152302, −6.66413902296177698957089246806, −5.67832781002998429403441765829, −5.16296511657495689373863409054, −4.08586608337112018589326259123, −2.85577409699452078525149474351, −2.20943146850104180000624805886, −0.74042326125537727438094954709, 0.74042326125537727438094954709, 2.20943146850104180000624805886, 2.85577409699452078525149474351, 4.08586608337112018589326259123, 5.16296511657495689373863409054, 5.67832781002998429403441765829, 6.66413902296177698957089246806, 7.79519147296136066151975152302, 8.200904634741764565590055787780, 9.830204412266161579588226374633

Graph of the ZZ-function along the critical line