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Properties

Label 2-58800-1.1-c1-0-123
Degree 22
Conductor 5880058800
Sign −1-1
Analytic cond. 469.520469.520
Root an. cond. 21.668421.6684
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 3·13-s + 2·17-s + 19-s + 2·23-s − 27-s − 8·29-s − 8·31-s − 7·37-s + 3·39-s − 8·43-s + 10·47-s − 2·51-s + 14·53-s − 57-s + 10·59-s − 7·61-s − 5·67-s − 2·69-s + 12·71-s − 11·73-s + 7·79-s + 81-s − 14·83-s + 8·87-s + 6·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.832·13-s + 0.485·17-s + 0.229·19-s + 0.417·23-s − 0.192·27-s − 1.48·29-s − 1.43·31-s − 1.15·37-s + 0.480·39-s − 1.21·43-s + 1.45·47-s − 0.280·51-s + 1.92·53-s − 0.132·57-s + 1.30·59-s − 0.896·61-s − 0.610·67-s − 0.240·69-s + 1.42·71-s − 1.28·73-s + 0.787·79-s + 1/9·81-s − 1.53·83-s + 0.857·87-s + 0.635·89-s + ⋯

Functional equation

Λ(s)=(58800s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(58800s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5880058800    =    24⋅3⋅52⋅722^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 469.520469.520
Root analytic conductor: 21.668421.6684
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 58800, ( :1/2), −1)(2,\ 58800,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
5 1 1
7 1 1
good11 1+pT2 1 + p T^{2}
13 1+3T+pT2 1 + 3 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−T+pT2 1 - T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+7T+pT2 1 + 7 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−10T+pT2 1 - 10 T + p T^{2}
53 1−14T+pT2 1 - 14 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1+5T+pT2 1 + 5 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+11T+pT2 1 + 11 T + p T^{2}
79 1−7T+pT2 1 - 7 T + p T^{2}
83 1+14T+pT2 1 + 14 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−9T+pT2 1 - 9 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

−14.73095881869097, −14.11010359047179, −13.48563855680084, −13.01073147544035, −12.55784375885712, −11.95606460752729, −11.66460038622333, −11.03556040787214, −10.47582464103819, −10.10341614037798, −9.474009790933225, −8.969359411998086, −8.468400622195011, −7.530938947503383, −7.304054301098625, −6.877608487984404, −5.988369322020594, −5.483497737244619, −5.178940100921644, −4.436608359285816, −3.712615342902426, −3.269312523620163, −2.270305694903136, −1.775759372634011, −0.8171657289212251, 0, 0.8171657289212251, 1.775759372634011, 2.270305694903136, 3.269312523620163, 3.712615342902426, 4.436608359285816, 5.178940100921644, 5.483497737244619, 5.988369322020594, 6.877608487984404, 7.304054301098625, 7.530938947503383, 8.468400622195011, 8.969359411998086, 9.474009790933225, 10.10341614037798, 10.47582464103819, 11.03556040787214, 11.66460038622333, 11.95606460752729, 12.55784375885712, 13.01073147544035, 13.48563855680084, 14.11010359047179, 14.73095881869097

Graph of the ZZ-function along the critical line