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Properties

Label 2-20400-1.1-c1-0-0
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 − 4·7-s + 9-s − 2·11-s − 4·13-s − 17-s + 4·19-s + 4·21-s − 8·23-s − 27-s + 8·29-s + 2·33-s + 2·37-s + 4·39-s − 4·41-s + 6·43-s − 12·47-s + 9·49-s + 51-s − 14·53-s − 4·57-s + 2·61-s − 4·63-s + 2·67-s + 8·69-s − 14·71-s − 2·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.242·17-s + 0.917·19-s + 0.872·21-s − 1.66·23-s − 0.192·27-s + 1.48·29-s + 0.348·33-s + 0.328·37-s + 0.640·39-s − 0.624·41-s + 0.914·43-s − 1.75·47-s + 9/7·49-s + 0.140·51-s − 1.92·53-s − 0.529·57-s + 0.256·61-s − 0.503·63-s + 0.244·67-s + 0.963·69-s − 1.66·71-s − 0.234·73-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 0.22475901750.2247590175
L(12)L(\frac12) ≈\approx 0.22475901750.2247590175
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+4T+pT2 1 + 4 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+4T+pT2 1 + 4 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 1−8T+pT2 1 - 8 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+4T+pT2 1 + 4 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1+12T+pT2 1 + 12 T + p T^{2}
53 1+14T+pT2 1 + 14 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1+14T+pT2 1 + 14 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+4T+pT2 1 + 4 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 1−14T+pT2 1 - 14 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

−15.74204111044190, −15.40121817556934, −14.35860461554662, −14.08966941878661, −13.32333537108345, −12.74723077228054, −12.50180380993533, −11.76012347883085, −11.44734186008884, −10.35433152129622, −10.12279999553137, −9.722503101562773, −9.112378762261052, −8.208574120059607, −7.642721122702005, −7.032792475571812, −6.356844325615584, −6.026421099815404, −5.192973525779386, −4.646266511578271, −3.857767917107552, −3.001942483920584, −2.581867221389900, −1.454822480901571, −0.1995120269963112, 0.1995120269963112, 1.454822480901571, 2.581867221389900, 3.001942483920584, 3.857767917107552, 4.646266511578271, 5.192973525779386, 6.026421099815404, 6.356844325615584, 7.032792475571812, 7.642721122702005, 8.208574120059607, 9.112378762261052, 9.722503101562773, 10.12279999553137, 10.35433152129622, 11.44734186008884, 11.76012347883085, 12.50180380993533, 12.74723077228054, 13.32333537108345, 14.08966941878661, 14.35860461554662, 15.40121817556934, 15.74204111044190

Graph of the ZZ-function along the critical line