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Properties

Label 2-4080-1.1-c1-0-25
Degree 22
Conductor 40804080
Sign 11
Analytic cond. 32.578932.5789
Root an. cond. 5.707795.70779
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 + 5-s + 3·7-s + 9-s − 3·11-s − 4·13-s + 15-s − 17-s − 19-s + 3·21-s + 4·23-s + 25-s + 27-s − 3·29-s + 8·31-s − 3·33-s + 3·35-s + 3·37-s − 4·39-s + 5·41-s + 8·43-s + 45-s + 9·47-s + 2·49-s − 51-s − 7·53-s − 3·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s + 0.258·15-s − 0.242·17-s − 0.229·19-s + 0.654·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.557·29-s + 1.43·31-s − 0.522·33-s + 0.507·35-s + 0.493·37-s − 0.640·39-s + 0.780·41-s + 1.21·43-s + 0.149·45-s + 1.31·47-s + 2/7·49-s − 0.140·51-s − 0.961·53-s − 0.404·55-s + ⋯

Functional equation

Λ(s)=(4080s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4080s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40804080    =    24⋅3⋅5⋅172^{4} \cdot 3 \cdot 5 \cdot 17
Sign: 11
Analytic conductor: 32.578932.5789
Root analytic conductor: 5.707795.70779
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4080, ( :1/2), 1)(2,\ 4080,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.8213296802.821329680
L(12)L(\frac12) ≈\approx 2.8213296802.821329680
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−T 1 - T
17 1+T 1 + T
good7 1−3T+pT2 1 - 3 T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
19 1+T+pT2 1 + T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−3T+pT2 1 - 3 T + p T^{2}
41 1−5T+pT2 1 - 5 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1−9T+pT2 1 - 9 T + p T^{2}
53 1+7T+pT2 1 + 7 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1+2T+pT2 1 + 2 T + p T^{2}
71 1−16T+pT2 1 - 16 T + p T^{2}
73 1−9T+pT2 1 - 9 T + p T^{2}
79 1−14T+pT2 1 - 14 T + p T^{2}
83 1−8T+pT2 1 - 8 T + p T^{2}
89 1−2T+pT2 1 - 2 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

−8.260382618805256542430323786308, −7.83053371110021056834400314726, −7.17357628311824753957788925827, −6.24898778293344730746320092815, −5.17145210780284634244296896801, −4.85692765750778214058282261269, −3.88935816953954937855088642106, −2.53416210932782266200703139162, −2.31191132427133956795448473973, −0.952314738010447951173545884501, 0.952314738010447951173545884501, 2.31191132427133956795448473973, 2.53416210932782266200703139162, 3.88935816953954937855088642106, 4.85692765750778214058282261269, 5.17145210780284634244296896801, 6.24898778293344730746320092815, 7.17357628311824753957788925827, 7.83053371110021056834400314726, 8.260382618805256542430323786308

Graph of the ZZ-function along the critical line