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Properties

Label 2-52e2-1.1-c1-0-3
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  − 1.69·3-s + 1.19·5-s − 4.29·7-s − 0.137·9-s − 3.69·11-s − 2.02·15-s − 7.85·17-s + 4.65·19-s + 7.26·21-s − 3.71·23-s − 3.56·25-s + 5.30·27-s + 4.29·29-s − 2.91·31-s + 6.24·33-s − 5.14·35-s − 8.03·37-s + 8.10·41-s − 0.884·43-s − 0.164·45-s + 4.58·47-s + 11.4·49-s + 13.2·51-s + 5.43·53-s − 4.42·55-s − 7.87·57-s − 2.32·59-s + ⋯
L(s)  = 1  − 0.976·3-s + 0.535·5-s − 1.62·7-s − 0.0456·9-s − 1.11·11-s − 0.523·15-s − 1.90·17-s + 1.06·19-s + 1.58·21-s − 0.774·23-s − 0.712·25-s + 1.02·27-s + 0.797·29-s − 0.522·31-s + 1.08·33-s − 0.869·35-s − 1.32·37-s + 1.26·41-s − 0.134·43-s − 0.0244·45-s + 0.668·47-s + 1.63·49-s + 1.86·51-s + 0.746·53-s − 0.596·55-s − 1.04·57-s − 0.303·59-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 0.46471254100.4647125410
L(12)L(\frac12) ≈\approx 0.46471254100.4647125410
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+1.69T+3T2 1 + 1.69T + 3T^{2}
5 1−1.19T+5T2 1 - 1.19T + 5T^{2}
7 1+4.29T+7T2 1 + 4.29T + 7T^{2}
11 1+3.69T+11T2 1 + 3.69T + 11T^{2}
17 1+7.85T+17T2 1 + 7.85T + 17T^{2}
19 1−4.65T+19T2 1 - 4.65T + 19T^{2}
23 1+3.71T+23T2 1 + 3.71T + 23T^{2}
29 1−4.29T+29T2 1 - 4.29T + 29T^{2}
31 1+2.91T+31T2 1 + 2.91T + 31T^{2}
37 1+8.03T+37T2 1 + 8.03T + 37T^{2}
41 1−8.10T+41T2 1 - 8.10T + 41T^{2}
43 1+0.884T+43T2 1 + 0.884T + 43T^{2}
47 1−4.58T+47T2 1 - 4.58T + 47T^{2}
53 1−5.43T+53T2 1 - 5.43T + 53T^{2}
59 1+2.32T+59T2 1 + 2.32T + 59T^{2}
61 1+6.25T+61T2 1 + 6.25T + 61T^{2}
67 1−3.33T+67T2 1 - 3.33T + 67T^{2}
71 1−4.35T+71T2 1 - 4.35T + 71T^{2}
73 1−3.82T+73T2 1 - 3.82T + 73T^{2}
79 1−10.5T+79T2 1 - 10.5T + 79T^{2}
83 1+5.24T+83T2 1 + 5.24T + 83T^{2}
89 1−9.46T+89T2 1 - 9.46T + 89T^{2}
97 1−3.67T+97T2 1 - 3.67T + 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

−9.091338015546489537130091087006, −8.050959708839000786269604100093, −6.99776496310863435472079497796, −6.46182042634360261582873910082, −5.77182700118171215171156998211, −5.23039556361652102578464774545, −4.13373328174842081229521486464, −3.02288058373235829804022087277, −2.23108081560296737513201880713, −0.41414188101732677903154585741, 0.41414188101732677903154585741, 2.23108081560296737513201880713, 3.02288058373235829804022087277, 4.13373328174842081229521486464, 5.23039556361652102578464774545, 5.77182700118171215171156998211, 6.46182042634360261582873910082, 6.99776496310863435472079497796, 8.050959708839000786269604100093, 9.091338015546489537130091087006

Graph of the ZZ-function along the critical line