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Properties

Label 2-4704-1.1-c1-0-36
Degree 22
Conductor 47044704
Sign 11
Analytic cond. 37.561637.5616
Root an. cond. 6.128756.12875
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  + 3-s + 2·5-s + 9-s + 2·11-s + 2·15-s + 2·17-s − 2·23-s − 25-s + 27-s + 6·29-s + 4·31-s + 2·33-s + 6·37-s − 2·41-s + 2·45-s + 2·51-s − 6·53-s + 4·55-s − 12·59-s + 12·61-s + 12·67-s − 2·69-s + 10·71-s + 12·73-s − 75-s − 12·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 1/3·9-s + 0.603·11-s + 0.516·15-s + 0.485·17-s − 0.417·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.348·33-s + 0.986·37-s − 0.312·41-s + 0.298·45-s + 0.280·51-s − 0.824·53-s + 0.539·55-s − 1.56·59-s + 1.53·61-s + 1.46·67-s − 0.240·69-s + 1.18·71-s + 1.40·73-s − 0.115·75-s − 1.35·79-s + 1/9·81-s + ⋯

Functional equation

Λ(s)=(4704s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4704s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47044704    =    25⋅3⋅722^{5} \cdot 3 \cdot 7^{2}
Sign: 11
Analytic conductor: 37.561637.5616
Root analytic conductor: 6.128756.12875
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4704, ( :1/2), 1)(2,\ 4704,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.2226158293.222615829
L(12)L(\frac12) ≈\approx 3.2226158293.222615829
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 1
3 1−T 1 - T
7 1 1
good5 1−2T+pT2 1 - 2 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+2T+pT2 1 + 2 T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
31 1−4T+pT2 1 - 4 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1−12T+pT2 1 - 12 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1−12T+pT2 1 - 12 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−14T+pT2 1 - 14 T + p T^{2}
97 1+12T+pT2 1 + 12 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

−8.286888431596555286657334298365, −7.73854178489723702308111376486, −6.69894450238348765055660509469, −6.25757774983507607022752153906, −5.39842320766302478198171451779, −4.54420676214985832644687837204, −3.70337928799270777592208975099, −2.79368202791990232736895950901, −1.98810207865713462948946984742, −1.02538988745536652107482241574, 1.02538988745536652107482241574, 1.98810207865713462948946984742, 2.79368202791990232736895950901, 3.70337928799270777592208975099, 4.54420676214985832644687837204, 5.39842320766302478198171451779, 6.25757774983507607022752153906, 6.69894450238348765055660509469, 7.73854178489723702308111376486, 8.286888431596555286657334298365

Graph of the ZZ-function along the critical line