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Properties

Label 2-52e2-1.1-c1-0-27
Degree 22
Conductor 27042704
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.56·3-s − 3.56·5-s + 2.56·7-s + 3.56·9-s − 2.56·11-s + 9.12·15-s − 5·17-s + 2.56·19-s − 6.56·21-s + 3.68·23-s + 7.68·25-s − 1.43·27-s − 5·29-s + 8·31-s + 6.56·33-s − 9.12·35-s − 37-s + 9.24·41-s + 6.56·43-s − 12.6·45-s − 4·47-s − 0.438·49-s + 12.8·51-s + 4.43·53-s + 9.12·55-s − 6.56·57-s + 2.56·59-s + ⋯
L(s)  = 1  − 1.47·3-s − 1.59·5-s + 0.968·7-s + 1.18·9-s − 0.772·11-s + 2.35·15-s − 1.21·17-s + 0.587·19-s − 1.43·21-s + 0.768·23-s + 1.53·25-s − 0.276·27-s − 0.928·29-s + 1.43·31-s + 1.14·33-s − 1.54·35-s − 0.164·37-s + 1.44·41-s + 1.00·43-s − 1.89·45-s − 0.583·47-s − 0.0626·49-s + 1.79·51-s + 0.609·53-s + 1.23·55-s − 0.869·57-s + 0.333·59-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: −1-1
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: 11
Selberg data: (2, 2704, ( :1/2), −1)(2,\ 2704,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+2.56T+3T2 1 + 2.56T + 3T^{2}
5 1+3.56T+5T2 1 + 3.56T + 5T^{2}
7 1−2.56T+7T2 1 - 2.56T + 7T^{2}
11 1+2.56T+11T2 1 + 2.56T + 11T^{2}
17 1+5T+17T2 1 + 5T + 17T^{2}
19 1−2.56T+19T2 1 - 2.56T + 19T^{2}
23 1−3.68T+23T2 1 - 3.68T + 23T^{2}
29 1+5T+29T2 1 + 5T + 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1+T+37T2 1 + T + 37T^{2}
41 1−9.24T+41T2 1 - 9.24T + 41T^{2}
43 1−6.56T+43T2 1 - 6.56T + 43T^{2}
47 1+4T+47T2 1 + 4T + 47T^{2}
53 1−4.43T+53T2 1 - 4.43T + 53T^{2}
59 1−2.56T+59T2 1 - 2.56T + 59T^{2}
61 1+7.24T+61T2 1 + 7.24T + 61T^{2}
67 1−9.43T+67T2 1 - 9.43T + 67T^{2}
71 1+7.68T+71T2 1 + 7.68T + 71T^{2}
73 1+1.31T+73T2 1 + 1.31T + 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1+2.24T+83T2 1 + 2.24T + 83T^{2}
89 1+9.68T+89T2 1 + 9.68T + 89T^{2}
97 1−2.80T+97T2 1 - 2.80T + 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

−8.249169078943114380213894314905, −7.61040276273750634236284507171, −7.03401257858780383813167782016, −6.10174803572499210922584177084, −5.14161687077441170328338407462, −4.67257958353095434834115931714, −3.97507229509595608839018308800, −2.65890713602005390669964368898, −1.04989136537845278309854557685, 0, 1.04989136537845278309854557685, 2.65890713602005390669964368898, 3.97507229509595608839018308800, 4.67257958353095434834115931714, 5.14161687077441170328338407462, 6.10174803572499210922584177084, 7.03401257858780383813167782016, 7.61040276273750634236284507171, 8.249169078943114380213894314905

Graph of the ZZ-function along the critical line