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Properties

Label 2-5408-1.1-c1-0-72
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  + 2.41·3-s − 2.82·5-s + 4.41·7-s + 2.82·9-s + 3.24·11-s − 6.82·15-s − 5.82·17-s − 1.24·19-s + 10.6·21-s − 1.24·23-s + 3.00·25-s − 0.414·27-s + 8.65·29-s + 5.65·31-s + 7.82·33-s − 12.4·35-s + 7.48·37-s + 5.82·41-s − 4.07·43-s − 8·45-s + 6·47-s + 12.4·49-s − 14.0·51-s − 2.82·53-s − 9.17·55-s − 3·57-s − 1.24·59-s + ⋯
L(s)  = 1  + 1.39·3-s − 1.26·5-s + 1.66·7-s + 0.942·9-s + 0.977·11-s − 1.76·15-s − 1.41·17-s − 0.285·19-s + 2.32·21-s − 0.259·23-s + 0.600·25-s − 0.0797·27-s + 1.60·29-s + 1.01·31-s + 1.36·33-s − 2.11·35-s + 1.23·37-s + 0.910·41-s − 0.620·43-s − 1.19·45-s + 0.875·47-s + 1.78·49-s − 1.97·51-s − 0.388·53-s − 1.23·55-s − 0.397·57-s − 0.161·59-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 3.3850876133.385087613
L(12)L(\frac12) ≈\approx 3.3850876133.385087613
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 1−2.41T+3T2 1 - 2.41T + 3T^{2}
5 1+2.82T+5T2 1 + 2.82T + 5T^{2}
7 1−4.41T+7T2 1 - 4.41T + 7T^{2}
11 1−3.24T+11T2 1 - 3.24T + 11T^{2}
17 1+5.82T+17T2 1 + 5.82T + 17T^{2}
19 1+1.24T+19T2 1 + 1.24T + 19T^{2}
23 1+1.24T+23T2 1 + 1.24T + 23T^{2}
29 1−8.65T+29T2 1 - 8.65T + 29T^{2}
31 1−5.65T+31T2 1 - 5.65T + 31T^{2}
37 1−7.48T+37T2 1 - 7.48T + 37T^{2}
41 1−5.82T+41T2 1 - 5.82T + 41T^{2}
43 1+4.07T+43T2 1 + 4.07T + 43T^{2}
47 1−6T+47T2 1 - 6T + 47T^{2}
53 1+2.82T+53T2 1 + 2.82T + 53T^{2}
59 1+1.24T+59T2 1 + 1.24T + 59T^{2}
61 1−7T+61T2 1 - 7T + 61T^{2}
67 1−13.2T+67T2 1 - 13.2T + 67T^{2}
71 1+7.24T+71T2 1 + 7.24T + 71T^{2}
73 1+12.4T+73T2 1 + 12.4T + 73T^{2}
79 1−6T+79T2 1 - 6T + 79T^{2}
83 1−4T+83T2 1 - 4T + 83T^{2}
89 1−3.34T+89T2 1 - 3.34T + 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

−8.163002021581106903165300470532, −7.80260847911718824346767896848, −7.02670816124336143730025666275, −6.21158818162114220661713603136, −4.80414533850404788822905637947, −4.31453885098004132106142828463, −3.87484245944889210098693645298, −2.76205241891822037752605317653, −2.05182273091076588194428043791, −0.968780946651762960778269978958, 0.968780946651762960778269978958, 2.05182273091076588194428043791, 2.76205241891822037752605317653, 3.87484245944889210098693645298, 4.31453885098004132106142828463, 4.80414533850404788822905637947, 6.21158818162114220661713603136, 7.02670816124336143730025666275, 7.80260847911718824346767896848, 8.163002021581106903165300470532

Graph of the ZZ-function along the critical line