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Properties

Label 2-5408-1.1-c1-0-47
Degree 22
Conductor 54085408
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·3-s + 2.82·5-s + 1.58·7-s − 2.82·9-s − 5.24·11-s − 1.17·15-s − 0.171·17-s + 7.24·19-s − 0.656·21-s + 7.24·23-s + 3.00·25-s + 2.41·27-s − 2.65·29-s − 5.65·31-s + 2.17·33-s + 4.48·35-s − 9.48·37-s + 0.171·41-s + 10.0·43-s − 8.00·45-s + 6·47-s − 4.48·49-s + 0.0710·51-s + 2.82·53-s − 14.8·55-s − 2.99·57-s + 7.24·59-s + ⋯
L(s)  = 1  − 0.239·3-s + 1.26·5-s + 0.599·7-s − 0.942·9-s − 1.58·11-s − 0.302·15-s − 0.0416·17-s + 1.66·19-s − 0.143·21-s + 1.51·23-s + 0.600·25-s + 0.464·27-s − 0.493·29-s − 1.01·31-s + 0.378·33-s + 0.758·35-s − 1.55·37-s + 0.0267·41-s + 1.53·43-s − 1.19·45-s + 0.875·47-s − 0.640·49-s + 0.00995·51-s + 0.388·53-s − 1.99·55-s − 0.397·57-s + 0.942·59-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: 11
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: 00
Selberg data: (2, 5408, ( :1/2), 1)(2,\ 5408,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1351264032.135126403
L(12)L(\frac12) ≈\approx 2.1351264032.135126403
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+0.414T+3T2 1 + 0.414T + 3T^{2}
5 1−2.82T+5T2 1 - 2.82T + 5T^{2}
7 1−1.58T+7T2 1 - 1.58T + 7T^{2}
11 1+5.24T+11T2 1 + 5.24T + 11T^{2}
17 1+0.171T+17T2 1 + 0.171T + 17T^{2}
19 1−7.24T+19T2 1 - 7.24T + 19T^{2}
23 1−7.24T+23T2 1 - 7.24T + 23T^{2}
29 1+2.65T+29T2 1 + 2.65T + 29T^{2}
31 1+5.65T+31T2 1 + 5.65T + 31T^{2}
37 1+9.48T+37T2 1 + 9.48T + 37T^{2}
41 1−0.171T+41T2 1 - 0.171T + 41T^{2}
43 1−10.0T+43T2 1 - 10.0T + 43T^{2}
47 1−6T+47T2 1 - 6T + 47T^{2}
53 1−2.82T+53T2 1 - 2.82T + 53T^{2}
59 1−7.24T+59T2 1 - 7.24T + 59T^{2}
61 1−7T+61T2 1 - 7T + 61T^{2}
67 1−4.75T+67T2 1 - 4.75T + 67T^{2}
71 1−1.24T+71T2 1 - 1.24T + 71T^{2}
73 1−4.48T+73T2 1 - 4.48T + 73T^{2}
79 1−6T+79T2 1 - 6T + 79T^{2}
83 1−4T+83T2 1 - 4T + 83T^{2}
89 1−14.6T+89T2 1 - 14.6T + 89T^{2}
97 1−9T+97T2 1 - 9T + 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

−8.114993285590380313870147274227, −7.48768716252550825910324545140, −6.74815102104003767634057697284, −5.60232411800284578314719073859, −5.41867426713687951873961939835, −4.99160731173212377682020100826, −3.49568209900732479762338730748, −2.68208283939971958074764602206, −2.00228685098563934553756299625, −0.789429202330035618389570570369, 0.789429202330035618389570570369, 2.00228685098563934553756299625, 2.68208283939971958074764602206, 3.49568209900732479762338730748, 4.99160731173212377682020100826, 5.41867426713687951873961939835, 5.60232411800284578314719073859, 6.74815102104003767634057697284, 7.48768716252550825910324545140, 8.114993285590380313870147274227

Graph of the ZZ-function along the critical line