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Properties

Label 2-550-1.1-c5-0-29
Degree 22
Conductor 550550
Sign 11
Analytic cond. 88.211188.2111
Root an. cond. 9.392079.39207
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 − 3-s + 16·4-s − 4·6-s + 166·7-s + 64·8-s − 242·9-s − 121·11-s − 16·12-s − 692·13-s + 664·14-s + 256·16-s + 738·17-s − 968·18-s + 1.42e3·19-s − 166·21-s − 484·22-s + 1.77e3·23-s − 64·24-s − 2.76e3·26-s + 485·27-s + 2.65e3·28-s − 2.06e3·29-s + 6.24e3·31-s + 1.02e3·32-s + 121·33-s + 2.95e3·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.0641·3-s + 1/2·4-s − 0.0453·6-s + 1.28·7-s + 0.353·8-s − 0.995·9-s − 0.301·11-s − 0.0320·12-s − 1.13·13-s + 0.905·14-s + 1/4·16-s + 0.619·17-s − 0.704·18-s + 0.904·19-s − 0.0821·21-s − 0.213·22-s + 0.701·23-s − 0.0226·24-s − 0.803·26-s + 0.128·27-s + 0.640·28-s − 0.455·29-s + 1.16·31-s + 0.176·32-s + 0.0193·33-s + 0.437·34-s + ⋯

Functional equation

Λ(s)=(550s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(550s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 550550    =    2⋅52⋅112 \cdot 5^{2} \cdot 11
Sign: 11
Analytic conductor: 88.211188.2111
Root analytic conductor: 9.392079.39207
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 550, ( :5/2), 1)(2,\ 550,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 3.7310469513.731046951
L(12)L(\frac12) ≈\approx 3.7310469513.731046951
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
5 1 1
11 1+p2T 1 + p^{2} T
good3 1+T+p5T2 1 + T + p^{5} T^{2}
7 1−166T+p5T2 1 - 166 T + p^{5} T^{2}
13 1+692T+p5T2 1 + 692 T + p^{5} T^{2}
17 1−738T+p5T2 1 - 738 T + p^{5} T^{2}
19 1−1424T+p5T2 1 - 1424 T + p^{5} T^{2}
23 1−1779T+p5T2 1 - 1779 T + p^{5} T^{2}
29 1+2064T+p5T2 1 + 2064 T + p^{5} T^{2}
31 1−6245T+p5T2 1 - 6245 T + p^{5} T^{2}
37 1−14785T+p5T2 1 - 14785 T + p^{5} T^{2}
41 1−5304T+p5T2 1 - 5304 T + p^{5} T^{2}
43 1+17798T+p5T2 1 + 17798 T + p^{5} T^{2}
47 1−17184T+p5T2 1 - 17184 T + p^{5} T^{2}
53 1−30726T+p5T2 1 - 30726 T + p^{5} T^{2}
59 1+34989T+p5T2 1 + 34989 T + p^{5} T^{2}
61 1+45940T+p5T2 1 + 45940 T + p^{5} T^{2}
67 1+25343T+p5T2 1 + 25343 T + p^{5} T^{2}
71 1−13311T+p5T2 1 - 13311 T + p^{5} T^{2}
73 1−53260T+p5T2 1 - 53260 T + p^{5} T^{2}
79 1−77234T+p5T2 1 - 77234 T + p^{5} T^{2}
83 1+55014T+p5T2 1 + 55014 T + p^{5} T^{2}
89 1−125415T+p5T2 1 - 125415 T + p^{5} T^{2}
97 1−88807T+p5T2 1 - 88807 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

−10.18486205571573001144578552174, −9.091048974704884358525046387742, −7.934444779330416257946216950292, −7.47640586989177705551622705709, −6.09115383407124629642224867023, −5.18654590582980070154820346850, −4.64104221004948157056163185344, −3.16377640370239447569317314717, −2.26328043286869344087498674374, −0.873869996874479234352790333521, 0.873869996874479234352790333521, 2.26328043286869344087498674374, 3.16377640370239447569317314717, 4.64104221004948157056163185344, 5.18654590582980070154820346850, 6.09115383407124629642224867023, 7.47640586989177705551622705709, 7.934444779330416257946216950292, 9.091048974704884358525046387742, 10.18486205571573001144578552174

Graph of the ZZ-function along the critical line