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Properties

Label 2-20400-1.1-c1-0-41
Degree 22
Conductor 2040020400
Sign 11
Analytic cond. 162.894162.894
Root an. cond. 12.763012.7630
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 + 3·7-s + 9-s + 2·11-s + 5·13-s + 17-s + 7·19-s − 3·21-s − 4·23-s − 27-s − 4·29-s − 5·31-s − 2·33-s − 6·37-s − 5·39-s + 2·41-s + 5·43-s + 12·47-s + 2·49-s − 51-s − 6·53-s − 7·57-s − 10·59-s + 13·61-s + 3·63-s + 5·67-s + 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.603·11-s + 1.38·13-s + 0.242·17-s + 1.60·19-s − 0.654·21-s − 0.834·23-s − 0.192·27-s − 0.742·29-s − 0.898·31-s − 0.348·33-s − 0.986·37-s − 0.800·39-s + 0.312·41-s + 0.762·43-s + 1.75·47-s + 2/7·49-s − 0.140·51-s − 0.824·53-s − 0.927·57-s − 1.30·59-s + 1.66·61-s + 0.377·63-s + 0.610·67-s + 0.481·69-s + ⋯

Functional equation

Λ(s)=(20400s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 20400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(20400s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 20400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2040020400    =    24⋅3⋅52⋅172^{4} \cdot 3 \cdot 5^{2} \cdot 17
Sign: 11
Analytic conductor: 162.894162.894
Root analytic conductor: 12.763012.7630
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 20400, ( :1/2), 1)(2,\ 20400,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.7209528762.720952876
L(12)L(\frac12) ≈\approx 2.7209528762.720952876
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
17 1−T 1 - T
good7 1−3T+pT2 1 - 3 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
19 1−7T+pT2 1 - 7 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1+5T+pT2 1 + 5 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 1−5T+pT2 1 - 5 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+10T+pT2 1 + 10 T + p T^{2}
61 1−13T+pT2 1 - 13 T + p T^{2}
67 1−5T+pT2 1 - 5 T + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1−10T+pT2 1 - 10 T + p T^{2}
97 1+T+pT2 1 + 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

−15.83829357338400, −15.15221967469137, −14.40715555359242, −13.99052838166346, −13.67430160336645, −12.79340392598051, −12.23930514959581, −11.71835870182537, −11.17525222679920, −10.91886163834379, −10.20689991848970, −9.391083324224935, −9.037263673099344, −8.208764890496551, −7.740198531375486, −7.145936343792368, −6.426012129517545, −5.584289436269852, −5.487259704726333, −4.548749997664092, −3.846240364705613, −3.385265903035296, −2.101670571451908, −1.437970370769223, −0.7792807896602335, 0.7792807896602335, 1.437970370769223, 2.101670571451908, 3.385265903035296, 3.846240364705613, 4.548749997664092, 5.487259704726333, 5.584289436269852, 6.426012129517545, 7.145936343792368, 7.740198531375486, 8.208764890496551, 9.037263673099344, 9.391083324224935, 10.20689991848970, 10.91886163834379, 11.17525222679920, 11.71835870182537, 12.23930514959581, 12.79340392598051, 13.67430160336645, 13.99052838166346, 14.40715555359242, 15.15221967469137, 15.83829357338400

Graph of the ZZ-function along the critical line