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Properties

Label 2-10608-1.1-c1-0-3
Degree 22
Conductor 1060810608
Sign 11
Analytic cond. 84.705384.7053
Root an. cond. 9.203549.20354
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 − 2·7-s + 9-s + 2·11-s − 13-s + 17-s + 4·19-s + 2·21-s + 8·23-s − 5·25-s − 27-s + 2·29-s + 6·31-s − 2·33-s + 8·37-s + 39-s + 4·41-s + 4·43-s − 3·49-s − 51-s − 14·53-s − 4·57-s − 8·59-s − 10·61-s − 2·63-s + 12·67-s − 8·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.242·17-s + 0.917·19-s + 0.436·21-s + 1.66·23-s − 25-s − 0.192·27-s + 0.371·29-s + 1.07·31-s − 0.348·33-s + 1.31·37-s + 0.160·39-s + 0.624·41-s + 0.609·43-s − 3/7·49-s − 0.140·51-s − 1.92·53-s − 0.529·57-s − 1.04·59-s − 1.28·61-s − 0.251·63-s + 1.46·67-s − 0.963·69-s + ⋯

Functional equation

Λ(s)=(10608s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(10608s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1060810608    =    24⋅3⋅13⋅172^{4} \cdot 3 \cdot 13 \cdot 17
Sign: 11
Analytic conductor: 84.705384.7053
Root analytic conductor: 9.203549.20354
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 10608, ( :1/2), 1)(2,\ 10608,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5906171401.590617140
L(12)L(\frac12) ≈\approx 1.5906171401.590617140
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
13 1+T 1 + T
17 1−T 1 - T
good5 1+pT2 1 + p T^{2}
7 1+2T+pT2 1 + 2 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1−6T+pT2 1 - 6 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1−4T+pT2 1 - 4 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+14T+pT2 1 + 14 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1+8T+pT2 1 + 8 T + p T^{2}
79 1+4T+pT2 1 + 4 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−2T+pT2 1 - 2 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

−16.67950205935192, −15.83628012328202, −15.69874036144325, −14.88175732306430, −14.22641144445315, −13.68498875595087, −12.99412920314538, −12.50959095501372, −11.96181754874469, −11.28343724918531, −10.91174657813054, −9.955552564402685, −9.557494843827964, −9.152308043756880, −8.107553014669859, −7.534767970266213, −6.819476911770268, −6.260982387527504, −5.705911463754020, −4.837393583908609, −4.286837482071561, −3.281202761502914, −2.770727786647186, −1.471974421445127, −0.6492996418946811, 0.6492996418946811, 1.471974421445127, 2.770727786647186, 3.281202761502914, 4.286837482071561, 4.837393583908609, 5.705911463754020, 6.260982387527504, 6.819476911770268, 7.534767970266213, 8.107553014669859, 9.152308043756880, 9.557494843827964, 9.955552564402685, 10.91174657813054, 11.28343724918531, 11.96181754874469, 12.50959095501372, 12.99412920314538, 13.68498875595087, 14.22641144445315, 14.88175732306430, 15.69874036144325, 15.83628012328202, 16.67950205935192

Graph of the ZZ-function along the critical line