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Properties

Label 2-8550-1.1-c1-0-89
Degree 22
Conductor 85508550
Sign 11
Analytic cond. 68.272068.2720
Root an. cond. 8.262698.26269
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5·7-s + 8-s + 4·11-s + 13-s + 5·14-s + 16-s − 3·17-s + 19-s + 4·22-s + 7·23-s + 26-s + 5·28-s + 3·29-s − 2·31-s + 32-s − 3·34-s + 2·37-s + 38-s + 6·41-s − 6·43-s + 4·44-s + 7·46-s + 18·49-s + 52-s − 13·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.88·7-s + 0.353·8-s + 1.20·11-s + 0.277·13-s + 1.33·14-s + 1/4·16-s − 0.727·17-s + 0.229·19-s + 0.852·22-s + 1.45·23-s + 0.196·26-s + 0.944·28-s + 0.557·29-s − 0.359·31-s + 0.176·32-s − 0.514·34-s + 0.328·37-s + 0.162·38-s + 0.937·41-s − 0.914·43-s + 0.603·44-s + 1.03·46-s + 18/7·49-s + 0.138·52-s − 1.78·53-s + ⋯

Functional equation

Λ(s)=(8550s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(8550s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 85508550    =    2⋅32⋅52⋅192 \cdot 3^{2} \cdot 5^{2} \cdot 19
Sign: 11
Analytic conductor: 68.272068.2720
Root analytic conductor: 8.262698.26269
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8550, ( :1/2), 1)(2,\ 8550,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.1360911025.136091102
L(12)L(\frac12) ≈\approx 5.1360911025.136091102
L(32)L(\frac{3}{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−T 1 - T
3 1 1
5 1 1
19 1−T 1 - T
good7 1−5T+pT2 1 - 5 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
23 1−7T+pT2 1 - 7 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+13T+pT2 1 + 13 T + p T^{2}
59 1−9T+pT2 1 - 9 T + p T^{2}
61 1+12T+pT2 1 + 12 T + p T^{2}
67 1−3T+pT2 1 - 3 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+11T+pT2 1 + 11 T + p T^{2}
79 1+2T+pT2 1 + 2 T + p T^{2}
83 1+10T+pT2 1 + 10 T + p T^{2}
89 1+2T+pT2 1 + 2 T + p T^{2}
97 1−2T+pT2 1 - 2 T + p 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

−7.64412675368342072100402643347, −7.07704541117290869170938897697, −6.35430569173332358701577738897, −5.60181065616457415736945006525, −4.73764288830413259853760333013, −4.52726401175650390589229310145, −3.63706267746930921054569078234, −2.68059787783471348034578632155, −1.68490358819888880085777025199, −1.14285782158350026604192915140, 1.14285782158350026604192915140, 1.68490358819888880085777025199, 2.68059787783471348034578632155, 3.63706267746930921054569078234, 4.52726401175650390589229310145, 4.73764288830413259853760333013, 5.60181065616457415736945006525, 6.35430569173332358701577738897, 7.07704541117290869170938897697, 7.64412675368342072100402643347

Graph of the ZZ-function along the critical line