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Properties

Label 2-52e2-1.1-c1-0-0
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  − 0.386·3-s − 3.17·5-s − 3.44·7-s − 2.85·9-s − 3.44·11-s + 1.22·15-s − 1.77·17-s − 3.33·19-s + 1.33·21-s + 0.386·23-s + 5.07·25-s + 2.26·27-s − 4.57·29-s − 11.0·31-s + 1.33·33-s + 10.9·35-s + 1.62·37-s − 2.26·41-s − 10.7·43-s + 9.04·45-s − 8.11·47-s + 4.85·49-s + 0.685·51-s + 11.6·53-s + 10.9·55-s + 1.28·57-s + 6.32·59-s + ⋯
L(s)  = 1  − 0.223·3-s − 1.41·5-s − 1.30·7-s − 0.950·9-s − 1.03·11-s + 0.316·15-s − 0.430·17-s − 0.764·19-s + 0.290·21-s + 0.0805·23-s + 1.01·25-s + 0.435·27-s − 0.849·29-s − 1.98·31-s + 0.231·33-s + 1.84·35-s + 0.266·37-s − 0.354·41-s − 1.63·43-s + 1.34·45-s − 1.18·47-s + 0.692·49-s + 0.0959·51-s + 1.60·53-s + 1.47·55-s + 0.170·57-s + 0.823·59-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 0.085340234570.08534023457
L(12)L(\frac12) ≈\approx 0.085340234570.08534023457
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+0.386T+3T2 1 + 0.386T + 3T^{2}
5 1+3.17T+5T2 1 + 3.17T + 5T^{2}
7 1+3.44T+7T2 1 + 3.44T + 7T^{2}
11 1+3.44T+11T2 1 + 3.44T + 11T^{2}
17 1+1.77T+17T2 1 + 1.77T + 17T^{2}
19 1+3.33T+19T2 1 + 3.33T + 19T^{2}
23 1−0.386T+23T2 1 - 0.386T + 23T^{2}
29 1+4.57T+29T2 1 + 4.57T + 29T^{2}
31 1+11.0T+31T2 1 + 11.0T + 31T^{2}
37 1−1.62T+37T2 1 - 1.62T + 37T^{2}
41 1+2.26T+41T2 1 + 2.26T + 41T^{2}
43 1+10.7T+43T2 1 + 10.7T + 43T^{2}
47 1+8.11T+47T2 1 + 8.11T + 47T^{2}
53 1−11.6T+53T2 1 - 11.6T + 53T^{2}
59 1−6.32T+59T2 1 - 6.32T + 59T^{2}
61 1−2.42T+61T2 1 - 2.42T + 61T^{2}
67 1−9.14T+67T2 1 - 9.14T + 67T^{2}
71 1+10.2T+71T2 1 + 10.2T + 71T^{2}
73 1−8.40T+73T2 1 - 8.40T + 73T^{2}
79 1+8.22T+79T2 1 + 8.22T + 79T^{2}
83 1+1.11T+83T2 1 + 1.11T + 83T^{2}
89 1−17.8T+89T2 1 - 17.8T + 89T^{2}
97 1+4.48T+97T2 1 + 4.48T + 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

−8.667614965565705896428426054603, −8.158039274458161539329821621107, −7.27530984160419824275845599678, −6.66894779916955244575736862128, −5.72831592479153046398081590413, −4.98340819878695095957373530243, −3.81020402386030550702407536809, −3.34239416422248359006747810188, −2.32986820345926741717739082766, −0.16842961856690561817906113073, 0.16842961856690561817906113073, 2.32986820345926741717739082766, 3.34239416422248359006747810188, 3.81020402386030550702407536809, 4.98340819878695095957373530243, 5.72831592479153046398081590413, 6.66894779916955244575736862128, 7.27530984160419824275845599678, 8.158039274458161539329821621107, 8.667614965565705896428426054603

Graph of the ZZ-function along the critical line