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Properties

Label 2-20400-1.1-c1-0-13
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 − 7-s + 9-s + 2·11-s − 3·13-s + 17-s − 19-s + 21-s + 8·23-s − 27-s − 31-s − 2·33-s − 6·37-s + 3·39-s − 10·41-s − 11·43-s + 8·47-s − 6·49-s − 51-s + 14·53-s + 57-s + 6·59-s − 7·61-s − 63-s + 13·67-s − 8·69-s − 2·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.832·13-s + 0.242·17-s − 0.229·19-s + 0.218·21-s + 1.66·23-s − 0.192·27-s − 0.179·31-s − 0.348·33-s − 0.986·37-s + 0.480·39-s − 1.56·41-s − 1.67·43-s + 1.16·47-s − 6/7·49-s − 0.140·51-s + 1.92·53-s + 0.132·57-s + 0.781·59-s − 0.896·61-s − 0.125·63-s + 1.58·67-s − 0.963·69-s − 0.237·71-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 1.3458243461.345824346
L(12)L(\frac12) ≈\approx 1.3458243461.345824346
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+T+pT2 1 + T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1+3T+pT2 1 + 3 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+T+pT2 1 + T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1+10T+pT2 1 + 10 T + p T^{2}
43 1+11T+pT2 1 + 11 T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−14T+pT2 1 - 14 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1−13T+pT2 1 - 13 T + p T^{2}
71 1+2T+pT2 1 + 2 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+16T+pT2 1 + 16 T + p T^{2}
83 1−16T+pT2 1 - 16 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+5T+pT2 1 + 5 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

−15.56520702164704, −15.07778786000399, −14.71933071344758, −13.97692683789056, −13.33079695037199, −12.93941310788690, −12.17537417501068, −11.93176258657281, −11.26899056620342, −10.68191849297530, −9.994037704139852, −9.742547251242481, −8.742121593440645, −8.616029683265745, −7.451005959619711, −7.033616081156645, −6.602994754550581, −5.839986743479305, −5.073951204973522, −4.803744060446337, −3.758532039668260, −3.266713098101324, −2.318557215678215, −1.454286631647516, −0.5097698497579768, 0.5097698497579768, 1.454286631647516, 2.318557215678215, 3.266713098101324, 3.758532039668260, 4.803744060446337, 5.073951204973522, 5.839986743479305, 6.602994754550581, 7.033616081156645, 7.451005959619711, 8.616029683265745, 8.742121593440645, 9.742547251242481, 9.994037704139852, 10.68191849297530, 11.26899056620342, 11.93176258657281, 12.17537417501068, 12.93941310788690, 13.33079695037199, 13.97692683789056, 14.71933071344758, 15.07778786000399, 15.56520702164704

Graph of the ZZ-function along the critical line