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Properties

Label 2-396-1.1-c5-0-6
Degree 22
Conductor 396396
Sign 11
Analytic cond. 63.511963.5119
Root an. cond. 7.969447.96944
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  − 33.5·5-s + 67.1·7-s + 121·11-s + 504.·13-s − 984.·17-s + 281.·19-s − 359.·23-s − 1.99e3·25-s + 5.02e3·29-s − 7.01e3·31-s − 2.25e3·35-s − 5.24e3·37-s + 1.38e4·41-s + 2.01e4·43-s − 6.78e3·47-s − 1.22e4·49-s + 2.72e4·53-s − 4.05e3·55-s − 1.90e4·59-s + 2.40e4·61-s − 1.69e4·65-s + 5.32e4·67-s + 4.42e4·71-s + 2.19e4·73-s + 8.12e3·77-s − 2.63e4·79-s + 1.94e4·83-s + ⋯
L(s)  = 1  − 0.600·5-s + 0.518·7-s + 0.301·11-s + 0.828·13-s − 0.826·17-s + 0.178·19-s − 0.141·23-s − 0.639·25-s + 1.10·29-s − 1.31·31-s − 0.310·35-s − 0.629·37-s + 1.28·41-s + 1.66·43-s − 0.447·47-s − 0.731·49-s + 1.33·53-s − 0.180·55-s − 0.714·59-s + 0.827·61-s − 0.496·65-s + 1.44·67-s + 1.04·71-s + 0.481·73-s + 0.156·77-s − 0.475·79-s + 0.309·83-s + ⋯

Functional equation

Λ(s)=(396s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(396s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 396396    =    22⋅32⋅112^{2} \cdot 3^{2} \cdot 11
Sign: 11
Analytic conductor: 63.511963.5119
Root analytic conductor: 7.969447.96944
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 396, ( :5/2), 1)(2,\ 396,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 1.9345883701.934588370
L(12)L(\frac12) ≈\approx 1.9345883701.934588370
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 1
11 1−121T 1 - 121T
good5 1+33.5T+3.12e3T2 1 + 33.5T + 3.12e3T^{2}
7 1−67.1T+1.68e4T2 1 - 67.1T + 1.68e4T^{2}
13 1−504.T+3.71e5T2 1 - 504.T + 3.71e5T^{2}
17 1+984.T+1.41e6T2 1 + 984.T + 1.41e6T^{2}
19 1−281.T+2.47e6T2 1 - 281.T + 2.47e6T^{2}
23 1+359.T+6.43e6T2 1 + 359.T + 6.43e6T^{2}
29 1−5.02e3T+2.05e7T2 1 - 5.02e3T + 2.05e7T^{2}
31 1+7.01e3T+2.86e7T2 1 + 7.01e3T + 2.86e7T^{2}
37 1+5.24e3T+6.93e7T2 1 + 5.24e3T + 6.93e7T^{2}
41 1−1.38e4T+1.15e8T2 1 - 1.38e4T + 1.15e8T^{2}
43 1−2.01e4T+1.47e8T2 1 - 2.01e4T + 1.47e8T^{2}
47 1+6.78e3T+2.29e8T2 1 + 6.78e3T + 2.29e8T^{2}
53 1−2.72e4T+4.18e8T2 1 - 2.72e4T + 4.18e8T^{2}
59 1+1.90e4T+7.14e8T2 1 + 1.90e4T + 7.14e8T^{2}
61 1−2.40e4T+8.44e8T2 1 - 2.40e4T + 8.44e8T^{2}
67 1−5.32e4T+1.35e9T2 1 - 5.32e4T + 1.35e9T^{2}
71 1−4.42e4T+1.80e9T2 1 - 4.42e4T + 1.80e9T^{2}
73 1−2.19e4T+2.07e9T2 1 - 2.19e4T + 2.07e9T^{2}
79 1+2.63e4T+3.07e9T2 1 + 2.63e4T + 3.07e9T^{2}
83 1−1.94e4T+3.93e9T2 1 - 1.94e4T + 3.93e9T^{2}
89 1−3.13e4T+5.58e9T2 1 - 3.13e4T + 5.58e9T^{2}
97 1−1.34e5T+8.58e9T2 1 - 1.34e5T + 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

−10.73237771126989262526609621461, −9.444563667083650407927268892953, −8.575805838877039743851838944940, −7.76969573023446593815273655828, −6.75840766658514869916072316675, −5.67643691692985461560033926788, −4.44864318704261173955482499609, −3.60357999334401235192699182660, −2.09907183802404112960617002281, −0.74625772058557356819582919855, 0.74625772058557356819582919855, 2.09907183802404112960617002281, 3.60357999334401235192699182660, 4.44864318704261173955482499609, 5.67643691692985461560033926788, 6.75840766658514869916072316675, 7.76969573023446593815273655828, 8.575805838877039743851838944940, 9.444563667083650407927268892953, 10.73237771126989262526609621461

Graph of the ZZ-function along the critical line