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Properties

Label 2-5408-1.1-c1-0-93
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  + 2.90·3-s + 2·5-s − 0.779·7-s + 5.46·9-s + 5.03·11-s + 5.81·15-s − 6.46·17-s + 0.779·19-s − 2.26·21-s + 7.16·23-s − 25-s + 7.16·27-s + 3·29-s + 1.55·31-s + 14.6·33-s − 1.55·35-s + 4.26·37-s − 3.19·41-s − 8.72·43-s + 10.9·45-s + 7.37·47-s − 6.39·49-s − 18.8·51-s + 9.46·53-s + 10.0·55-s + 2.26·57-s − 12.4·59-s + ⋯
L(s)  = 1  + 1.67·3-s + 0.894·5-s − 0.294·7-s + 1.82·9-s + 1.51·11-s + 1.50·15-s − 1.56·17-s + 0.178·19-s − 0.494·21-s + 1.49·23-s − 0.200·25-s + 1.37·27-s + 0.557·29-s + 0.280·31-s + 2.55·33-s − 0.263·35-s + 0.701·37-s − 0.499·41-s − 1.33·43-s + 1.62·45-s + 1.07·47-s − 0.913·49-s − 2.63·51-s + 1.29·53-s + 1.35·55-s + 0.300·57-s − 1.61·59-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 4.8841696244.884169624
L(12)L(\frac12) ≈\approx 4.8841696244.884169624
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 1−2.90T+3T2 1 - 2.90T + 3T^{2}
5 1−2T+5T2 1 - 2T + 5T^{2}
7 1+0.779T+7T2 1 + 0.779T + 7T^{2}
11 1−5.03T+11T2 1 - 5.03T + 11T^{2}
17 1+6.46T+17T2 1 + 6.46T + 17T^{2}
19 1−0.779T+19T2 1 - 0.779T + 19T^{2}
23 1−7.16T+23T2 1 - 7.16T + 23T^{2}
29 1−3T+29T2 1 - 3T + 29T^{2}
31 1−1.55T+31T2 1 - 1.55T + 31T^{2}
37 1−4.26T+37T2 1 - 4.26T + 37T^{2}
41 1+3.19T+41T2 1 + 3.19T + 41T^{2}
43 1+8.72T+43T2 1 + 8.72T + 43T^{2}
47 1−7.37T+47T2 1 - 7.37T + 47T^{2}
53 1−9.46T+53T2 1 - 9.46T + 53T^{2}
59 1+12.4T+59T2 1 + 12.4T + 59T^{2}
61 1−3T+61T2 1 - 3T + 61T^{2}
67 1−0.779T+67T2 1 - 0.779T + 67T^{2}
71 1−15.1T+71T2 1 - 15.1T + 71T^{2}
73 1−6T+73T2 1 - 6T + 73T^{2}
79 1+10.0T+79T2 1 + 10.0T + 79T^{2}
83 1−10.0T+83T2 1 - 10.0T + 83T^{2}
89 1−9.19T+89T2 1 - 9.19T + 89T^{2}
97 1−10.2T+97T2 1 - 10.2T + 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

−8.460695621636765761500106931762, −7.50262596352735598612383420848, −6.61205385110675913215127124042, −6.47256619802580000826229932857, −5.12677623494920046051003888956, −4.26664682453302570013059360108, −3.58056082701344505131959704587, −2.74533127673538015364014662278, −2.05611678814104365367289862196, −1.21248115921947163502407719856, 1.21248115921947163502407719856, 2.05611678814104365367289862196, 2.74533127673538015364014662278, 3.58056082701344505131959704587, 4.26664682453302570013059360108, 5.12677623494920046051003888956, 6.47256619802580000826229932857, 6.61205385110675913215127124042, 7.50262596352735598612383420848, 8.460695621636765761500106931762

Graph of the ZZ-function along the critical line