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Properties

Label 2-4704-1.1-c1-0-26
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 + 4·5-s + 9-s − 2·11-s + 2·13-s − 4·15-s + 4·19-s − 6·23-s + 11·25-s − 27-s − 10·29-s + 8·31-s + 2·33-s + 10·37-s − 2·39-s + 4·41-s − 8·43-s + 4·45-s + 4·47-s + 10·53-s − 8·55-s − 4·57-s − 8·59-s + 6·61-s + 8·65-s + 4·67-s + 6·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.78·5-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 1.03·15-s + 0.917·19-s − 1.25·23-s + 11/5·25-s − 0.192·27-s − 1.85·29-s + 1.43·31-s + 0.348·33-s + 1.64·37-s − 0.320·39-s + 0.624·41-s − 1.21·43-s + 0.596·45-s + 0.583·47-s + 1.37·53-s − 1.07·55-s − 0.529·57-s − 1.04·59-s + 0.768·61-s + 0.992·65-s + 0.488·67-s + 0.722·69-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 2.3538823032.353882303
L(12)L(\frac12) ≈\approx 2.3538823032.353882303
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 1−4T+pT2 1 - 4 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1+10T+pT2 1 + 10 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−4T+pT2 1 - 4 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1−14T+pT2 1 - 14 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+4T+pT2 1 + 4 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

−8.279041445583954269580152420931, −7.53628213226894683621941986320, −6.59168190460477202975948009550, −5.97431604276500309005136289819, −5.57054459148342379924700545698, −4.86263496822252259250448584118, −3.80178865956147297903992698888, −2.63860153156949338353954958777, −1.91683092707172386565079248122, −0.904574473973546245991803949373, 0.904574473973546245991803949373, 1.91683092707172386565079248122, 2.63860153156949338353954958777, 3.80178865956147297903992698888, 4.86263496822252259250448584118, 5.57054459148342379924700545698, 5.97431604276500309005136289819, 6.59168190460477202975948009550, 7.53628213226894683621941986320, 8.279041445583954269580152420931

Graph of the ZZ-function along the critical line