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Properties

Label 2-5376-1.1-c1-0-4
Degree 22
Conductor 53765376
Sign 11
Analytic cond. 42.927542.9275
Root an. cond. 6.551916.55191
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-s − 4.21·5-s + 7-s + 9-s − 5.57·11-s − 1.35·13-s − 4.21·15-s − 4.86·17-s + 2.64·19-s + 21-s − 2.86·23-s + 12.7·25-s + 27-s − 2.64·29-s − 1.35·31-s − 5.57·33-s − 4.21·35-s − 5.79·37-s − 1.35·39-s + 0.425·41-s + 8.43·43-s − 4.21·45-s + 11.1·47-s + 49-s − 4.86·51-s − 5.79·53-s + 23.5·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.88·5-s + 0.377·7-s + 0.333·9-s − 1.68·11-s − 0.376·13-s − 1.08·15-s − 1.17·17-s + 0.606·19-s + 0.218·21-s − 0.596·23-s + 2.55·25-s + 0.192·27-s − 0.490·29-s − 0.243·31-s − 0.970·33-s − 0.713·35-s − 0.952·37-s − 0.217·39-s + 0.0663·41-s + 1.28·43-s − 0.628·45-s + 1.62·47-s + 0.142·49-s − 0.680·51-s − 0.795·53-s + 3.17·55-s + ⋯

Functional equation

Λ(s)=(5376s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5376s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 53765376    =    28⋅3⋅72^{8} \cdot 3 \cdot 7
Sign: 11
Analytic conductor: 42.927542.9275
Root analytic conductor: 6.551916.55191
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5376, ( :1/2), 1)(2,\ 5376,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.86974740460.8697474046
L(12)L(\frac12) ≈\approx 0.86974740460.8697474046
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
3 1−T 1 - T
7 1−T 1 - T
good5 1+4.21T+5T2 1 + 4.21T + 5T^{2}
11 1+5.57T+11T2 1 + 5.57T + 11T^{2}
13 1+1.35T+13T2 1 + 1.35T + 13T^{2}
17 1+4.86T+17T2 1 + 4.86T + 17T^{2}
19 1−2.64T+19T2 1 - 2.64T + 19T^{2}
23 1+2.86T+23T2 1 + 2.86T + 23T^{2}
29 1+2.64T+29T2 1 + 2.64T + 29T^{2}
31 1+1.35T+31T2 1 + 1.35T + 31T^{2}
37 1+5.79T+37T2 1 + 5.79T + 37T^{2}
41 1−0.425T+41T2 1 - 0.425T + 41T^{2}
43 1−8.43T+43T2 1 - 8.43T + 43T^{2}
47 1−11.1T+47T2 1 - 11.1T + 47T^{2}
53 1+5.79T+53T2 1 + 5.79T + 53T^{2}
59 1+9.72T+59T2 1 + 9.72T + 59T^{2}
61 1+15.0T+61T2 1 + 15.0T + 61T^{2}
67 1−8T+67T2 1 - 8T + 67T^{2}
71 1+14.0T+71T2 1 + 14.0T + 71T^{2}
73 1−1.56T+73T2 1 - 1.56T + 73T^{2}
79 1−0.436T+79T2 1 - 0.436T + 79T^{2}
83 1−12.8T+83T2 1 - 12.8T + 83T^{2}
89 1−15.5T+89T2 1 - 15.5T + 89T^{2}
97 1−9.14T+97T2 1 - 9.14T + 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

−7.997133171861454623564215165468, −7.49072714701430459571510315956, −7.31383728113386947948984938495, −6.03217638130432095758600373888, −4.91335883820452824524937189607, −4.53740689263379891385435165636, −3.65534848475923715054724500159, −2.94397358118513562923130696792, −2.08339744454559113826202527035, −0.46154861675791770967768661077, 0.46154861675791770967768661077, 2.08339744454559113826202527035, 2.94397358118513562923130696792, 3.65534848475923715054724500159, 4.53740689263379891385435165636, 4.91335883820452824524937189607, 6.03217638130432095758600373888, 7.31383728113386947948984938495, 7.49072714701430459571510315956, 7.997133171861454623564215165468

Graph of the ZZ-function along the critical line