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Properties

Label 2-5408-1.1-c1-0-150
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  + 1.73·3-s + 3.46·5-s − 3·7-s − 5·11-s + 5.99·15-s + 7·17-s − 5·19-s − 5.19·21-s − 5.19·23-s + 6.99·25-s − 5.19·27-s − 5·29-s − 2·31-s − 8.66·33-s − 10.3·35-s − 5.19·37-s − 1.73·41-s − 5.19·43-s + 4·47-s + 2·49-s + 12.1·51-s + 4·53-s − 17.3·55-s − 8.66·57-s − 7·59-s + 3·61-s + ⋯
L(s)  = 1  + 1.00·3-s + 1.54·5-s − 1.13·7-s − 1.50·11-s + 1.54·15-s + 1.69·17-s − 1.14·19-s − 1.13·21-s − 1.08·23-s + 1.39·25-s − 1.00·27-s − 0.928·29-s − 0.359·31-s − 1.50·33-s − 1.75·35-s − 0.854·37-s − 0.270·41-s − 0.792·43-s + 0.583·47-s + 0.285·49-s + 1.69·51-s + 0.549·53-s − 2.33·55-s − 1.14·57-s − 0.911·59-s + 0.384·61-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 1−1.73T+3T2 1 - 1.73T + 3T^{2}
5 1−3.46T+5T2 1 - 3.46T + 5T^{2}
7 1+3T+7T2 1 + 3T + 7T^{2}
11 1+5T+11T2 1 + 5T + 11T^{2}
17 1−7T+17T2 1 - 7T + 17T^{2}
19 1+5T+19T2 1 + 5T + 19T^{2}
23 1+5.19T+23T2 1 + 5.19T + 23T^{2}
29 1+5T+29T2 1 + 5T + 29T^{2}
31 1+2T+31T2 1 + 2T + 31T^{2}
37 1+5.19T+37T2 1 + 5.19T + 37T^{2}
41 1+1.73T+41T2 1 + 1.73T + 41T^{2}
43 1+5.19T+43T2 1 + 5.19T + 43T^{2}
47 1−4T+47T2 1 - 4T + 47T^{2}
53 1−4T+53T2 1 - 4T + 53T^{2}
59 1+7T+59T2 1 + 7T + 59T^{2}
61 1−3T+61T2 1 - 3T + 61T^{2}
67 1−3T+67T2 1 - 3T + 67T^{2}
71 1+7T+71T2 1 + 7T + 71T^{2}
73 1+3.46T+73T2 1 + 3.46T + 73T^{2}
79 1−3.46T+79T2 1 - 3.46T + 79T^{2}
83 1−14T+83T2 1 - 14T + 83T^{2}
89 1+1.73T+89T2 1 + 1.73T + 89T^{2}
97 1+8.66T+97T2 1 + 8.66T + 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

−7.937985447532870769502708935694, −7.18555194996690999610474164184, −6.20036243603643610623908836777, −5.72714835900501338984956196883, −5.16255748131257740172559120024, −3.77023152874057563501130278683, −3.09529316933892911448125077077, −2.41555511615693334495499462860, −1.76752275129678328409173856115, 0, 1.76752275129678328409173856115, 2.41555511615693334495499462860, 3.09529316933892911448125077077, 3.77023152874057563501130278683, 5.16255748131257740172559120024, 5.72714835900501338984956196883, 6.20036243603643610623908836777, 7.18555194996690999610474164184, 7.937985447532870769502708935694

Graph of the ZZ-function along the critical line