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Properties

Label 2-96-1.1-c5-0-5
Degree 22
Conductor 9696
Sign 11
Analytic cond. 15.396815.3968
Root an. cond. 3.923883.92388
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s + 107.·5-s + 149.·7-s + 81·9-s − 434.·11-s − 392.·13-s + 963.·15-s + 803.·17-s + 854.·19-s + 1.34e3·21-s − 4.59e3·23-s + 8.34e3·25-s + 729·27-s + 6.79e3·29-s − 4.79e3·31-s − 3.91e3·33-s + 1.59e4·35-s − 909.·37-s − 3.53e3·39-s − 2.37e3·41-s − 8.66e3·43-s + 8.67e3·45-s + 1.68e4·47-s + 5.41e3·49-s + 7.23e3·51-s + 1.08e4·53-s − 4.65e4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.91·5-s + 1.14·7-s + 0.333·9-s − 1.08·11-s − 0.644·13-s + 1.10·15-s + 0.674·17-s + 0.543·19-s + 0.663·21-s − 1.81·23-s + 2.66·25-s + 0.192·27-s + 1.50·29-s − 0.896·31-s − 0.625·33-s + 2.20·35-s − 0.109·37-s − 0.371·39-s − 0.220·41-s − 0.714·43-s + 0.638·45-s + 1.11·47-s + 0.322·49-s + 0.389·51-s + 0.529·53-s − 2.07·55-s + ⋯

Functional equation

Λ(s)=(96s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(96s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 9696    =    25⋅32^{5} \cdot 3
Sign: 11
Analytic conductor: 15.396815.3968
Root analytic conductor: 3.923883.92388
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 96, ( :5/2), 1)(2,\ 96,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 3.2072769843.207276984
L(12)L(\frac12) ≈\approx 3.2072769843.207276984
L(72)L(\frac{7}{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 1−9T 1 - 9T
good5 1−107.T+3.12e3T2 1 - 107.T + 3.12e3T^{2}
7 1−149.T+1.68e4T2 1 - 149.T + 1.68e4T^{2}
11 1+434.T+1.61e5T2 1 + 434.T + 1.61e5T^{2}
13 1+392.T+3.71e5T2 1 + 392.T + 3.71e5T^{2}
17 1−803.T+1.41e6T2 1 - 803.T + 1.41e6T^{2}
19 1−854.T+2.47e6T2 1 - 854.T + 2.47e6T^{2}
23 1+4.59e3T+6.43e6T2 1 + 4.59e3T + 6.43e6T^{2}
29 1−6.79e3T+2.05e7T2 1 - 6.79e3T + 2.05e7T^{2}
31 1+4.79e3T+2.86e7T2 1 + 4.79e3T + 2.86e7T^{2}
37 1+909.T+6.93e7T2 1 + 909.T + 6.93e7T^{2}
41 1+2.37e3T+1.15e8T2 1 + 2.37e3T + 1.15e8T^{2}
43 1+8.66e3T+1.47e8T2 1 + 8.66e3T + 1.47e8T^{2}
47 1−1.68e4T+2.29e8T2 1 - 1.68e4T + 2.29e8T^{2}
53 1−1.08e4T+4.18e8T2 1 - 1.08e4T + 4.18e8T^{2}
59 1−8.30e3T+7.14e8T2 1 - 8.30e3T + 7.14e8T^{2}
61 1+3.59e4T+8.44e8T2 1 + 3.59e4T + 8.44e8T^{2}
67 1−2.50e4T+1.35e9T2 1 - 2.50e4T + 1.35e9T^{2}
71 1+5.78e4T+1.80e9T2 1 + 5.78e4T + 1.80e9T^{2}
73 1+6.80e4T+2.07e9T2 1 + 6.80e4T + 2.07e9T^{2}
79 1+9.87e4T+3.07e9T2 1 + 9.87e4T + 3.07e9T^{2}
83 1−3.48e4T+3.93e9T2 1 - 3.48e4T + 3.93e9T^{2}
89 1−3.46e4T+5.58e9T2 1 - 3.46e4T + 5.58e9T^{2}
97 1−3.98e4T+8.58e9T2 1 - 3.98e4T + 8.58e9T^{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

−13.31474362258981328664402008445, −12.10912481332933460893494155649, −10.42098276578784588577857983225, −9.913199136562547562677378010591, −8.626802183863672773537982939927, −7.50694669914831601736673824687, −5.84140299252974182624626090693, −4.90464331602273466484279221205, −2.61657655144675772129530448179, −1.60951289621691539682008742832, 1.60951289621691539682008742832, 2.61657655144675772129530448179, 4.90464331602273466484279221205, 5.84140299252974182624626090693, 7.50694669914831601736673824687, 8.626802183863672773537982939927, 9.913199136562547562677378010591, 10.42098276578784588577857983225, 12.10912481332933460893494155649, 13.31474362258981328664402008445

Graph of the ZZ-function along the critical line