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Properties

Label 2-5408-1.1-c1-0-76
Degree 22
Conductor 54085408
Sign 11
Analytic cond. 43.183043.1830
Root an. cond. 6.571386.57138
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  + 2·3-s + 5-s + 9-s + 4·11-s + 2·15-s + 3·17-s − 2·19-s + 2·23-s − 4·25-s − 4·27-s + 5·29-s + 2·31-s + 8·33-s + 5·37-s + 3·41-s − 4·43-s + 45-s + 6·47-s − 7·49-s + 6·51-s + 13·53-s + 4·55-s − 4·57-s + 12·59-s − 7·61-s − 14·67-s + 4·69-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.516·15-s + 0.727·17-s − 0.458·19-s + 0.417·23-s − 4/5·25-s − 0.769·27-s + 0.928·29-s + 0.359·31-s + 1.39·33-s + 0.821·37-s + 0.468·41-s − 0.609·43-s + 0.149·45-s + 0.875·47-s − 49-s + 0.840·51-s + 1.78·53-s + 0.539·55-s − 0.529·57-s + 1.56·59-s − 0.896·61-s − 1.71·67-s + 0.481·69-s + ⋯

Functional equation

Λ(s)=(5408s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5408s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 54085408    =    25⋅1322^{5} \cdot 13^{2}
Sign: 11
Analytic conductor: 43.183043.1830
Root analytic conductor: 6.571386.57138
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5408, ( :1/2), 1)(2,\ 5408,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.6773504913.677350491
L(12)L(\frac12) ≈\approx 3.6773504913.677350491
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
13 1 1
good3 1−2T+pT2 1 - 2 T + p T^{2}
5 1−T+pT2 1 - T + p T^{2}
7 1+pT2 1 + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
17 1−3T+pT2 1 - 3 T + p T^{2}
19 1+2T+pT2 1 + 2 T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1−5T+pT2 1 - 5 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−5T+pT2 1 - 5 T + p T^{2}
41 1−3T+pT2 1 - 3 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−13T+pT2 1 - 13 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1+7T+pT2 1 + 7 T + p T^{2}
67 1+14T+pT2 1 + 14 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−7T+pT2 1 - 7 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−14T+pT2 1 - 14 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−8.191185078182776739405830467558, −7.66844491577874889402472423892, −6.73258599174818484524749586897, −6.14237126427874163595995351602, −5.30036516143146284562099037082, −4.24308965441678315135450587683, −3.64707230410667219853785623773, −2.78765805022077372013361123276, −2.03552943509357525289762875452, −1.02783059228903960222011057272, 1.02783059228903960222011057272, 2.03552943509357525289762875452, 2.78765805022077372013361123276, 3.64707230410667219853785623773, 4.24308965441678315135450587683, 5.30036516143146284562099037082, 6.14237126427874163595995351602, 6.73258599174818484524749586897, 7.66844491577874889402472423892, 8.191185078182776739405830467558

Graph of the ZZ-function along the critical line