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Properties

Label 2-5445-1.1-c1-0-27
Degree 22
Conductor 54455445
Sign 11
Analytic cond. 43.478543.4785
Root an. cond. 6.593826.59382
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  − 2·4-s + 5-s − 7-s + 2·13-s + 4·16-s − 6·17-s − 7·19-s − 2·20-s + 6·23-s + 25-s + 2·28-s − 31-s − 35-s − 7·37-s + 6·41-s + 8·43-s − 6·49-s − 4·52-s + 6·53-s + 12·59-s − 61-s − 8·64-s + 2·65-s − 7·67-s + 12·68-s − 6·71-s − 13·73-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s − 0.377·7-s + 0.554·13-s + 16-s − 1.45·17-s − 1.60·19-s − 0.447·20-s + 1.25·23-s + 1/5·25-s + 0.377·28-s − 0.179·31-s − 0.169·35-s − 1.15·37-s + 0.937·41-s + 1.21·43-s − 6/7·49-s − 0.554·52-s + 0.824·53-s + 1.56·59-s − 0.128·61-s − 64-s + 0.248·65-s − 0.855·67-s + 1.45·68-s − 0.712·71-s − 1.52·73-s + ⋯

Functional equation

Λ(s)=(5445s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5445s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 54455445    =    32⋅5⋅1123^{2} \cdot 5 \cdot 11^{2}
Sign: 11
Analytic conductor: 43.478543.4785
Root analytic conductor: 6.593826.59382
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5445, ( :1/2), 1)(2,\ 5445,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2003079711.200307971
L(12)L(\frac12) ≈\approx 1.2003079711.200307971
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)
bad3 1 1
5 1−T 1 - T
11 1 1
good2 1+pT2 1 + p T^{2}
7 1+T+pT2 1 + T + 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+7T+pT2 1 + 7 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+T+pT2 1 + T + p T^{2}
37 1+7T+pT2 1 + 7 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+T+pT2 1 + T + p T^{2}
67 1+7T+pT2 1 + 7 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1+13T+pT2 1 + 13 T + p T^{2}
79 1−11T+pT2 1 - 11 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1−18T+pT2 1 - 18 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

−8.485453916851019492858569664820, −7.43711424473686269431816778687, −6.61280187235421899316911179550, −6.05793013905856029986241325619, −5.20779791692855593559951650094, −4.44793309957623836809852653633, −3.87307626331343778310808010397, −2.84635947998777897362121685781, −1.86441514423774169059856966061, −0.58620388806846202709441772114, 0.58620388806846202709441772114, 1.86441514423774169059856966061, 2.84635947998777897362121685781, 3.87307626331343778310808010397, 4.44793309957623836809852653633, 5.20779791692855593559951650094, 6.05793013905856029986241325619, 6.61280187235421899316911179550, 7.43711424473686269431816778687, 8.485453916851019492858569664820

Graph of the ZZ-function along the critical line