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Properties

Label 2-57-1.1-c9-0-1
Degree 22
Conductor 5757
Sign 11
Analytic cond. 29.357029.3570
Root an. cond. 5.418215.41821
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 14.1·2-s − 81·3-s − 313.·4-s − 1.99e3·5-s − 1.14e3·6-s + 5.78e3·7-s − 1.16e4·8-s + 6.56e3·9-s − 2.80e4·10-s − 5.53e4·11-s + 2.53e4·12-s − 1.79e5·13-s + 8.16e4·14-s + 1.61e5·15-s − 3.74e3·16-s + 4.08e5·17-s + 9.25e4·18-s + 1.30e5·19-s + 6.23e5·20-s − 4.68e5·21-s − 7.81e5·22-s + 2.42e6·23-s + 9.42e5·24-s + 2.01e6·25-s − 2.52e6·26-s − 5.31e5·27-s − 1.81e6·28-s + ⋯
L(s)  = 1  + 0.623·2-s − 0.577·3-s − 0.611·4-s − 1.42·5-s − 0.359·6-s + 0.910·7-s − 1.00·8-s + 0.333·9-s − 0.887·10-s − 1.14·11-s + 0.353·12-s − 1.73·13-s + 0.567·14-s + 0.822·15-s − 0.0142·16-s + 1.18·17-s + 0.207·18-s + 0.229·19-s + 0.871·20-s − 0.525·21-s − 0.710·22-s + 1.80·23-s + 0.579·24-s + 1.02·25-s − 1.08·26-s − 0.192·27-s − 0.557·28-s + ⋯

Functional equation

Λ(s)=(57s/2 ΓC(s) L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(57s/2 ΓC(s+9/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5757    =    3⋅193 \cdot 19
Sign: 11
Analytic conductor: 29.357029.3570
Root analytic conductor: 5.418215.41821
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 57, ( :9/2), 1)(2,\ 57,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 0.91609743620.9160974362
L(12)L(\frac12) ≈\approx 0.91609743620.9160974362
L(112)L(\frac{11}{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)
bad3 1+81T 1 + 81T
19 1−1.30e5T 1 - 1.30e5T
good2 1−14.1T+512T2 1 - 14.1T + 512T^{2}
5 1+1.99e3T+1.95e6T2 1 + 1.99e3T + 1.95e6T^{2}
7 1−5.78e3T+4.03e7T2 1 - 5.78e3T + 4.03e7T^{2}
11 1+5.53e4T+2.35e9T2 1 + 5.53e4T + 2.35e9T^{2}
13 1+1.79e5T+1.06e10T2 1 + 1.79e5T + 1.06e10T^{2}
17 1−4.08e5T+1.18e11T2 1 - 4.08e5T + 1.18e11T^{2}
23 1−2.42e6T+1.80e12T2 1 - 2.42e6T + 1.80e12T^{2}
29 1−7.26e5T+1.45e13T2 1 - 7.26e5T + 1.45e13T^{2}
31 1−1.22e6T+2.64e13T2 1 - 1.22e6T + 2.64e13T^{2}
37 1−1.00e7T+1.29e14T2 1 - 1.00e7T + 1.29e14T^{2}
41 1−4.11e6T+3.27e14T2 1 - 4.11e6T + 3.27e14T^{2}
43 1+4.14e7T+5.02e14T2 1 + 4.14e7T + 5.02e14T^{2}
47 1+3.13e7T+1.11e15T2 1 + 3.13e7T + 1.11e15T^{2}
53 1+1.07e7T+3.29e15T2 1 + 1.07e7T + 3.29e15T^{2}
59 1−1.65e8T+8.66e15T2 1 - 1.65e8T + 8.66e15T^{2}
61 1+1.04e7T+1.16e16T2 1 + 1.04e7T + 1.16e16T^{2}
67 1+6.39e7T+2.72e16T2 1 + 6.39e7T + 2.72e16T^{2}
71 1+2.64e8T+4.58e16T2 1 + 2.64e8T + 4.58e16T^{2}
73 1−2.14e8T+5.88e16T2 1 - 2.14e8T + 5.88e16T^{2}
79 1+2.10e8T+1.19e17T2 1 + 2.10e8T + 1.19e17T^{2}
83 1−3.55e8T+1.86e17T2 1 - 3.55e8T + 1.86e17T^{2}
89 1−4.88e8T+3.50e17T2 1 - 4.88e8T + 3.50e17T^{2}
97 1+5.97e8T+7.60e17T2 1 + 5.97e8T + 7.60e17T^{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

−13.05797600200556136503749635467, −12.15137625007035227542932250628, −11.38450510432387763910714125167, −9.948305263429299756979393855776, −8.217038594124342982049119801465, −7.35434734979526962443058828228, −5.16151561385744866670799514639, −4.71403577955576008349055984938, −3.11651361435607347901695697439, −0.55756096172790935185096040134, 0.55756096172790935185096040134, 3.11651361435607347901695697439, 4.71403577955576008349055984938, 5.16151561385744866670799514639, 7.35434734979526962443058828228, 8.217038594124342982049119801465, 9.948305263429299756979393855776, 11.38450510432387763910714125167, 12.15137625007035227542932250628, 13.05797600200556136503749635467

Graph of the ZZ-function along the critical line