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Properties

Label 2-1323-1.1-c3-0-116
Degree 22
Conductor 13231323
Sign 11
Analytic cond. 78.059578.0595
Root an. cond. 8.835138.83513
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.46·2-s + 21.8·4-s + 0.199·5-s + 75.5·8-s + 1.08·10-s + 28.4·11-s + 32.5·13-s + 237.·16-s − 115.·17-s + 21.1·19-s + 4.34·20-s + 155.·22-s + 93.7·23-s − 124.·25-s + 177.·26-s + 231.·29-s + 281.·31-s + 695.·32-s − 630.·34-s − 146.·37-s + 115.·38-s + 15.0·40-s − 111.·41-s + 392.·43-s + 620.·44-s + 512.·46-s + 273.·47-s + ⋯
L(s)  = 1  + 1.93·2-s + 2.72·4-s + 0.0178·5-s + 3.33·8-s + 0.0343·10-s + 0.779·11-s + 0.695·13-s + 3.71·16-s − 1.64·17-s + 0.255·19-s + 0.0486·20-s + 1.50·22-s + 0.850·23-s − 0.999·25-s + 1.34·26-s + 1.48·29-s + 1.63·31-s + 3.84·32-s − 3.18·34-s − 0.651·37-s + 0.493·38-s + 0.0594·40-s − 0.422·41-s + 1.39·43-s + 2.12·44-s + 1.64·46-s + 0.847·47-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 11
Analytic conductor: 78.059578.0595
Root analytic conductor: 8.835138.83513
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1323, ( :3/2), 1)(2,\ 1323,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 9.1191757789.119175778
L(12)L(\frac12) ≈\approx 9.1191757789.119175778
L(52)L(\frac{5}{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 1
7 1 1
good2 1−5.46T+8T2 1 - 5.46T + 8T^{2}
5 1−0.199T+125T2 1 - 0.199T + 125T^{2}
11 1−28.4T+1.33e3T2 1 - 28.4T + 1.33e3T^{2}
13 1−32.5T+2.19e3T2 1 - 32.5T + 2.19e3T^{2}
17 1+115.T+4.91e3T2 1 + 115.T + 4.91e3T^{2}
19 1−21.1T+6.85e3T2 1 - 21.1T + 6.85e3T^{2}
23 1−93.7T+1.21e4T2 1 - 93.7T + 1.21e4T^{2}
29 1−231.T+2.43e4T2 1 - 231.T + 2.43e4T^{2}
31 1−281.T+2.97e4T2 1 - 281.T + 2.97e4T^{2}
37 1+146.T+5.06e4T2 1 + 146.T + 5.06e4T^{2}
41 1+111.T+6.89e4T2 1 + 111.T + 6.89e4T^{2}
43 1−392.T+7.95e4T2 1 - 392.T + 7.95e4T^{2}
47 1−273.T+1.03e5T2 1 - 273.T + 1.03e5T^{2}
53 1+340.T+1.48e5T2 1 + 340.T + 1.48e5T^{2}
59 1+696.T+2.05e5T2 1 + 696.T + 2.05e5T^{2}
61 1+370.T+2.26e5T2 1 + 370.T + 2.26e5T^{2}
67 1−87.1T+3.00e5T2 1 - 87.1T + 3.00e5T^{2}
71 1+88.3T+3.57e5T2 1 + 88.3T + 3.57e5T^{2}
73 1+803.T+3.89e5T2 1 + 803.T + 3.89e5T^{2}
79 1−364.T+4.93e5T2 1 - 364.T + 4.93e5T^{2}
83 1−921.T+5.71e5T2 1 - 921.T + 5.71e5T^{2}
89 1+211.T+7.04e5T2 1 + 211.T + 7.04e5T^{2}
97 1−845.T+9.12e5T2 1 - 845.T + 9.12e5T^{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

−9.273671400522211943743129977485, −8.248025003291125509669939959750, −7.15700987497031083896530075198, −6.44055063108547925135865393245, −5.97685460593182160109825110399, −4.74656406976674737890711139423, −4.30504271645170610062690929716, −3.30956944547911110256672801830, −2.41493247290483231253852739880, −1.27134869292084255218831776704, 1.27134869292084255218831776704, 2.41493247290483231253852739880, 3.30956944547911110256672801830, 4.30504271645170610062690929716, 4.74656406976674737890711139423, 5.97685460593182160109825110399, 6.44055063108547925135865393245, 7.15700987497031083896530075198, 8.248025003291125509669939959750, 9.273671400522211943743129977485

Graph of the ZZ-function along the critical line