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Properties

Label 2-48-1.1-c27-0-13
Degree 22
Conductor 4848
Sign 11
Analytic cond. 221.690221.690
Root an. cond. 14.889214.8892
Motivic weight 2727
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.59e6·3-s + 4.36e9·5-s − 3.11e11·7-s + 2.54e12·9-s + 6.06e13·11-s + 6.46e14·13-s + 6.96e15·15-s + 2.78e16·17-s + 3.23e17·19-s − 4.96e17·21-s − 2.77e18·23-s + 1.16e19·25-s + 4.05e18·27-s − 4.40e18·29-s − 1.35e20·31-s + 9.66e19·33-s − 1.35e21·35-s + 2.42e21·37-s + 1.03e21·39-s − 1.00e22·41-s − 6.87e21·43-s + 1.11e22·45-s + 7.32e22·47-s + 3.11e22·49-s + 4.44e22·51-s + 2.25e23·53-s + 2.64e23·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.60·5-s − 1.21·7-s + 0.333·9-s + 0.529·11-s + 0.592·13-s + 0.924·15-s + 0.681·17-s + 1.76·19-s − 0.700·21-s − 1.14·23-s + 1.56·25-s + 0.192·27-s − 0.0797·29-s − 0.995·31-s + 0.305·33-s − 1.94·35-s + 1.63·37-s + 0.341·39-s − 1.70·41-s − 0.610·43-s + 0.533·45-s + 1.95·47-s + 0.473·49-s + 0.393·51-s + 1.18·53-s + 0.847·55-s + ⋯

Functional equation

Λ(s)=(48s/2 ΓC(s) L(s)=( Λ(28−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}
Λ(s)=(48s/2 ΓC(s+27/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+27/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4848    =    24⋅32^{4} \cdot 3
Sign: 11
Analytic conductor: 221.690221.690
Root analytic conductor: 14.889214.8892
Motivic weight: 2727
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 48, ( :27/2), 1)(2,\ 48,\ (\ :27/2),\ 1)

Particular Values

L(14)L(14) ≈\approx 4.3678317434.367831743
L(12)L(\frac12) ≈\approx 4.3678317434.367831743
L(292)L(\frac{29}{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
3 1−1.59e6T 1 - 1.59e6T
good5 1−4.36e9T+7.45e18T2 1 - 4.36e9T + 7.45e18T^{2}
7 1+3.11e11T+6.57e22T2 1 + 3.11e11T + 6.57e22T^{2}
11 1−6.06e13T+1.31e28T2 1 - 6.06e13T + 1.31e28T^{2}
13 1−6.46e14T+1.19e30T2 1 - 6.46e14T + 1.19e30T^{2}
17 1−2.78e16T+1.66e33T2 1 - 2.78e16T + 1.66e33T^{2}
19 1−3.23e17T+3.36e34T2 1 - 3.23e17T + 3.36e34T^{2}
23 1+2.77e18T+5.84e36T2 1 + 2.77e18T + 5.84e36T^{2}
29 1+4.40e18T+3.05e39T2 1 + 4.40e18T + 3.05e39T^{2}
31 1+1.35e20T+1.84e40T2 1 + 1.35e20T + 1.84e40T^{2}
37 1−2.42e21T+2.19e42T2 1 - 2.42e21T + 2.19e42T^{2}
41 1+1.00e22T+3.50e43T2 1 + 1.00e22T + 3.50e43T^{2}
43 1+6.87e21T+1.26e44T2 1 + 6.87e21T + 1.26e44T^{2}
47 1−7.32e22T+1.40e45T2 1 - 7.32e22T + 1.40e45T^{2}
53 1−2.25e23T+3.59e46T2 1 - 2.25e23T + 3.59e46T^{2}
59 1−6.81e22T+6.50e47T2 1 - 6.81e22T + 6.50e47T^{2}
61 1−2.06e24T+1.59e48T2 1 - 2.06e24T + 1.59e48T^{2}
67 1+5.18e24T+2.01e49T2 1 + 5.18e24T + 2.01e49T^{2}
71 1+9.12e24T+9.63e49T2 1 + 9.12e24T + 9.63e49T^{2}
73 1+6.99e24T+2.04e50T2 1 + 6.99e24T + 2.04e50T^{2}
79 1+3.46e25T+1.72e51T2 1 + 3.46e25T + 1.72e51T^{2}
83 1−5.24e25T+6.53e51T2 1 - 5.24e25T + 6.53e51T^{2}
89 1−2.05e26T+4.30e52T2 1 - 2.05e26T + 4.30e52T^{2}
97 1−3.43e26T+4.39e53T2 1 - 3.43e26T + 4.39e53T^{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

−10.16615095017602300676912623779, −9.690470117714450293190299982649, −8.865676492525466583574453318746, −7.30796439538500621827271816730, −6.18841066958626046258365738347, −5.50476369418244116681712293780, −3.75246637682478132245336427694, −2.88897734265127925217414309474, −1.80636949032861795819643728766, −0.865440736491405725008687668011, 0.865440736491405725008687668011, 1.80636949032861795819643728766, 2.88897734265127925217414309474, 3.75246637682478132245336427694, 5.50476369418244116681712293780, 6.18841066958626046258365738347, 7.30796439538500621827271816730, 8.865676492525466583574453318746, 9.690470117714450293190299982649, 10.16615095017602300676912623779

Graph of the ZZ-function along the critical line