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Properties

Label 2-5408-1.1-c1-0-55
Degree 22
Conductor 54085408
Sign −1-1
Analytic cond. 43.183043.1830
Root an. cond. 6.571386.57138
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.23·3-s − 2·5-s − 4.62·7-s − 1.46·9-s − 2.14·11-s + 2.47·15-s + 0.464·17-s + 4.62·19-s + 5.73·21-s + 5.53·23-s − 25-s + 5.53·27-s + 3·29-s + 9.25·31-s + 2.66·33-s + 9.25·35-s − 7.73·37-s − 7.19·41-s + 3.71·43-s + 2.92·45-s + 11.7·47-s + 14.3·49-s − 0.575·51-s + 2.53·53-s + 4.29·55-s − 5.73·57-s − 9.58·59-s + ⋯
L(s)  = 1  − 0.715·3-s − 0.894·5-s − 1.74·7-s − 0.488·9-s − 0.647·11-s + 0.639·15-s + 0.112·17-s + 1.06·19-s + 1.25·21-s + 1.15·23-s − 0.200·25-s + 1.06·27-s + 0.557·29-s + 1.66·31-s + 0.463·33-s + 1.56·35-s − 1.27·37-s − 1.12·41-s + 0.566·43-s + 0.436·45-s + 1.71·47-s + 2.05·49-s − 0.0805·51-s + 0.348·53-s + 0.578·55-s − 0.759·57-s − 1.24·59-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: −1-1
Analytic conductor: 43.183043.1830
Root analytic conductor: 6.571386.57138
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 5408, ( :1/2), −1)(2,\ 5408,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+1.23T+3T2 1 + 1.23T + 3T^{2}
5 1+2T+5T2 1 + 2T + 5T^{2}
7 1+4.62T+7T2 1 + 4.62T + 7T^{2}
11 1+2.14T+11T2 1 + 2.14T + 11T^{2}
17 1−0.464T+17T2 1 - 0.464T + 17T^{2}
19 1−4.62T+19T2 1 - 4.62T + 19T^{2}
23 1−5.53T+23T2 1 - 5.53T + 23T^{2}
29 1−3T+29T2 1 - 3T + 29T^{2}
31 1−9.25T+31T2 1 - 9.25T + 31T^{2}
37 1+7.73T+37T2 1 + 7.73T + 37T^{2}
41 1+7.19T+41T2 1 + 7.19T + 41T^{2}
43 1−3.71T+43T2 1 - 3.71T + 43T^{2}
47 1−11.7T+47T2 1 - 11.7T + 47T^{2}
53 1−2.53T+53T2 1 - 2.53T + 53T^{2}
59 1+9.58T+59T2 1 + 9.58T + 59T^{2}
61 1−3T+61T2 1 - 3T + 61T^{2}
67 1−4.62T+67T2 1 - 4.62T + 67T^{2}
71 1+6.43T+71T2 1 + 6.43T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1+4.29T+79T2 1 + 4.29T + 79T^{2}
83 1+4.29T+83T2 1 + 4.29T + 83T^{2}
89 1−1.19T+89T2 1 - 1.19T + 89T^{2}
97 1+13.7T+97T2 1 + 13.7T + 97T^{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

−7.67028055175067437080811384122, −7.01473896863702454587700602065, −6.41976004844199625543147128010, −5.66485689895821009192110380172, −5.03987297724527830968347051936, −4.05030222647526952038492905098, −3.11352139298563440182612829165, −2.80913507372797213764629690077, −0.866009157093505434128931064751, 0, 0.866009157093505434128931064751, 2.80913507372797213764629690077, 3.11352139298563440182612829165, 4.05030222647526952038492905098, 5.03987297724527830968347051936, 5.66485689895821009192110380172, 6.41976004844199625543147128010, 7.01473896863702454587700602065, 7.67028055175067437080811384122

Graph of the ZZ-function along the critical line