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Properties

Label 2-2400-1.1-c1-0-21
Degree 22
Conductor 24002400
Sign 11
Analytic cond. 19.164019.1640
Root an. cond. 4.377684.37768
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 + 2·13-s + 6·17-s + 4·21-s + 4·23-s + 27-s − 2·29-s − 8·31-s − 6·37-s + 2·39-s − 6·41-s − 12·43-s + 12·47-s + 9·49-s + 6·51-s + 10·53-s + 8·59-s − 10·61-s + 4·63-s + 12·67-s + 4·69-s + 8·71-s − 10·73-s + 16·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.554·13-s + 1.45·17-s + 0.872·21-s + 0.834·23-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 1.82·43-s + 1.75·47-s + 9/7·49-s + 0.840·51-s + 1.37·53-s + 1.04·59-s − 1.28·61-s + 0.503·63-s + 1.46·67-s + 0.481·69-s + 0.949·71-s − 1.17·73-s + 1.80·79-s + 1/9·81-s + ⋯

Functional equation

Λ(s)=(2400s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2400s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24002400    =    25⋅3⋅522^{5} \cdot 3 \cdot 5^{2}
Sign: 11
Analytic conductor: 19.164019.1640
Root analytic conductor: 4.377684.37768
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2400, ( :1/2), 1)(2,\ 2400,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.9167782652.916778265
L(12)L(\frac12) ≈\approx 2.9167782652.916778265
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
good7 1−4T+pT2 1 - 4 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1+12T+pT2 1 + 12 T + p T^{2}
47 1−12T+pT2 1 - 12 T + p T^{2}
53 1−10T+pT2 1 - 10 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+18T+pT2 1 + 18 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

−8.705123030777718551062740746738, −8.327275882404076664812739978493, −7.50997107238151608216676165196, −6.92530805420045658384192775774, −5.52811097399805640485701497969, −5.15807146102017226607398116108, −4.01150010595379173537756771542, −3.28316059994551513635790000973, −2.01099980219901421650957341119, −1.20740065177480422433876237100, 1.20740065177480422433876237100, 2.01099980219901421650957341119, 3.28316059994551513635790000973, 4.01150010595379173537756771542, 5.15807146102017226607398116108, 5.52811097399805640485701497969, 6.92530805420045658384192775774, 7.50997107238151608216676165196, 8.327275882404076664812739978493, 8.705123030777718551062740746738

Graph of the ZZ-function along the critical line