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Properties

Label 2-52e2-1.1-c1-0-4
Degree 22
Conductor 27042704
Sign 11
Analytic cond. 21.591521.5915
Root an. cond. 4.646674.64667
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  − 3.33·3-s − 0.932·5-s − 3.45·7-s + 8.09·9-s + 1.85·11-s + 3.10·15-s + 0.218·17-s + 5.10·19-s + 11.5·21-s − 7.49·23-s − 4.13·25-s − 16.9·27-s − 3.19·29-s − 7.91·31-s − 6.19·33-s + 3.22·35-s − 4.26·37-s − 3.27·41-s + 4.58·43-s − 7.55·45-s − 1.30·47-s + 4.95·49-s − 0.727·51-s − 2.68·53-s − 1.73·55-s − 17.0·57-s + 9.23·59-s + ⋯
L(s)  = 1  − 1.92·3-s − 0.417·5-s − 1.30·7-s + 2.69·9-s + 0.560·11-s + 0.802·15-s + 0.0529·17-s + 1.17·19-s + 2.51·21-s − 1.56·23-s − 0.826·25-s − 3.26·27-s − 0.593·29-s − 1.42·31-s − 1.07·33-s + 0.545·35-s − 0.701·37-s − 0.510·41-s + 0.699·43-s − 1.12·45-s − 0.191·47-s + 0.707·49-s − 0.101·51-s − 0.369·53-s − 0.233·55-s − 2.25·57-s + 1.20·59-s + ⋯

Functional equation

Λ(s)=(2704s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(2704s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27042704    =    24⋅1322^{4} \cdot 13^{2}
Sign: 11
Analytic conductor: 21.591521.5915
Root analytic conductor: 4.646674.64667
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2704, ( :1/2), 1)(2,\ 2704,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.37432076390.3743207639
L(12)L(\frac12) ≈\approx 0.37432076390.3743207639
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+3.33T+3T2 1 + 3.33T + 3T^{2}
5 1+0.932T+5T2 1 + 0.932T + 5T^{2}
7 1+3.45T+7T2 1 + 3.45T + 7T^{2}
11 1−1.85T+11T2 1 - 1.85T + 11T^{2}
17 1−0.218T+17T2 1 - 0.218T + 17T^{2}
19 1−5.10T+19T2 1 - 5.10T + 19T^{2}
23 1+7.49T+23T2 1 + 7.49T + 23T^{2}
29 1+3.19T+29T2 1 + 3.19T + 29T^{2}
31 1+7.91T+31T2 1 + 7.91T + 31T^{2}
37 1+4.26T+37T2 1 + 4.26T + 37T^{2}
41 1+3.27T+41T2 1 + 3.27T + 41T^{2}
43 1−4.58T+43T2 1 - 4.58T + 43T^{2}
47 1+1.30T+47T2 1 + 1.30T + 47T^{2}
53 1+2.68T+53T2 1 + 2.68T + 53T^{2}
59 1−9.23T+59T2 1 - 9.23T + 59T^{2}
61 1+2.22T+61T2 1 + 2.22T + 61T^{2}
67 1+4.43T+67T2 1 + 4.43T + 67T^{2}
71 1−2.60T+71T2 1 - 2.60T + 71T^{2}
73 1−3.28T+73T2 1 - 3.28T + 73T^{2}
79 1+7.33T+79T2 1 + 7.33T + 79T^{2}
83 1−3.58T+83T2 1 - 3.58T + 83T^{2}
89 1−5.07T+89T2 1 - 5.07T + 89T^{2}
97 1−13.4T+97T2 1 - 13.4T + 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

−9.134485689142110133124603630178, −7.69693110332998385789667527447, −7.14725959439029762244628315840, −6.37033319998848613884659009480, −5.84121451593483583917027359552, −5.18178379057667395719386365983, −4.03814686770283145865128472113, −3.55290001962095798739099436142, −1.74898532609017671213728075936, −0.41826711553313859668215894465, 0.41826711553313859668215894465, 1.74898532609017671213728075936, 3.55290001962095798739099436142, 4.03814686770283145865128472113, 5.18178379057667395719386365983, 5.84121451593483583917027359552, 6.37033319998848613884659009480, 7.14725959439029762244628315840, 7.69693110332998385789667527447, 9.134485689142110133124603630178

Graph of the ZZ-function along the critical line