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Properties

Label 2-40e2-1.1-c1-0-11
Degree 22
Conductor 16001600
Sign 11
Analytic cond. 12.776012.7760
Root an. cond. 3.574363.57436
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 − 2·9-s + 5·11-s + 5·17-s + 5·19-s + 2·21-s − 6·23-s − 5·27-s − 4·29-s − 10·31-s + 5·33-s + 10·37-s + 5·41-s + 4·43-s + 8·47-s − 3·49-s + 5·51-s + 10·53-s + 5·57-s + 10·61-s − 4·63-s + 3·67-s − 6·69-s − 5·73-s + 10·77-s − 10·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s − 2/3·9-s + 1.50·11-s + 1.21·17-s + 1.14·19-s + 0.436·21-s − 1.25·23-s − 0.962·27-s − 0.742·29-s − 1.79·31-s + 0.870·33-s + 1.64·37-s + 0.780·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.700·51-s + 1.37·53-s + 0.662·57-s + 1.28·61-s − 0.503·63-s + 0.366·67-s − 0.722·69-s − 0.585·73-s + 1.13·77-s − 1.12·79-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 11
Analytic conductor: 12.776012.7760
Root analytic conductor: 3.574363.57436
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1600, ( :1/2), 1)(2,\ 1600,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4205095712.420509571
L(12)L(\frac12) ≈\approx 2.4205095712.420509571
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
5 1 1
good3 1−T+pT2 1 - T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1−5T+pT2 1 - 5 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−5T+pT2 1 - 5 T + p T^{2}
19 1−5T+pT2 1 - 5 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1+10T+pT2 1 + 10 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−10T+pT2 1 - 10 T + p T^{2}
67 1−3T+pT2 1 - 3 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+5T+pT2 1 + 5 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1+T+pT2 1 + T + p T^{2}
89 1+9T+pT2 1 + 9 T + p T^{2}
97 1−10T+pT2 1 - 10 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

−9.323754252560504956925848056540, −8.683366516272630610807238108491, −7.76550095341300361088846412304, −7.32233528761629317808083018374, −5.94672002738926315494959415182, −5.49835151922875749387532575327, −4.11708760011461050263971856342, −3.52621022238586572141128158530, −2.29156961942451897610628689947, −1.16753716752208245348849419893, 1.16753716752208245348849419893, 2.29156961942451897610628689947, 3.52621022238586572141128158530, 4.11708760011461050263971856342, 5.49835151922875749387532575327, 5.94672002738926315494959415182, 7.32233528761629317808083018374, 7.76550095341300361088846412304, 8.683366516272630610807238108491, 9.323754252560504956925848056540

Graph of the ZZ-function along the critical line