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Properties

Label 2-1323-1.1-c3-0-109
Degree 22
Conductor 13231323
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.67·2-s + 13.8·4-s + 15.5·5-s − 27.2·8-s − 72.6·10-s − 63.4·11-s − 53.1·13-s + 16.7·16-s + 68.4·17-s + 86.0·19-s + 215.·20-s + 296.·22-s − 46.9·23-s + 116.·25-s + 248.·26-s + 169.·29-s + 141.·31-s + 139.·32-s − 319.·34-s − 411.·37-s − 402.·38-s − 423.·40-s + 49.0·41-s − 356.·43-s − 878.·44-s + 219.·46-s + 387.·47-s + ⋯
L(s)  = 1  − 1.65·2-s + 1.72·4-s + 1.39·5-s − 1.20·8-s − 2.29·10-s − 1.74·11-s − 1.13·13-s + 0.261·16-s + 0.976·17-s + 1.03·19-s + 2.40·20-s + 2.87·22-s − 0.425·23-s + 0.932·25-s + 1.87·26-s + 1.08·29-s + 0.821·31-s + 0.772·32-s − 1.61·34-s − 1.83·37-s − 1.71·38-s − 1.67·40-s + 0.186·41-s − 1.26·43-s − 3.00·44-s + 0.702·46-s + 1.20·47-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: −1-1
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: 11
Selberg data: (2, 1323, ( :3/2), −1)(2,\ 1323,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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+4.67T+8T2 1 + 4.67T + 8T^{2}
5 1−15.5T+125T2 1 - 15.5T + 125T^{2}
11 1+63.4T+1.33e3T2 1 + 63.4T + 1.33e3T^{2}
13 1+53.1T+2.19e3T2 1 + 53.1T + 2.19e3T^{2}
17 1−68.4T+4.91e3T2 1 - 68.4T + 4.91e3T^{2}
19 1−86.0T+6.85e3T2 1 - 86.0T + 6.85e3T^{2}
23 1+46.9T+1.21e4T2 1 + 46.9T + 1.21e4T^{2}
29 1−169.T+2.43e4T2 1 - 169.T + 2.43e4T^{2}
31 1−141.T+2.97e4T2 1 - 141.T + 2.97e4T^{2}
37 1+411.T+5.06e4T2 1 + 411.T + 5.06e4T^{2}
41 1−49.0T+6.89e4T2 1 - 49.0T + 6.89e4T^{2}
43 1+356.T+7.95e4T2 1 + 356.T + 7.95e4T^{2}
47 1−387.T+1.03e5T2 1 - 387.T + 1.03e5T^{2}
53 1−184.T+1.48e5T2 1 - 184.T + 1.48e5T^{2}
59 1−627.T+2.05e5T2 1 - 627.T + 2.05e5T^{2}
61 1+821.T+2.26e5T2 1 + 821.T + 2.26e5T^{2}
67 1+95.9T+3.00e5T2 1 + 95.9T + 3.00e5T^{2}
71 1+733.T+3.57e5T2 1 + 733.T + 3.57e5T^{2}
73 1−750.T+3.89e5T2 1 - 750.T + 3.89e5T^{2}
79 1−23.5T+4.93e5T2 1 - 23.5T + 4.93e5T^{2}
83 1−592.T+5.71e5T2 1 - 592.T + 5.71e5T^{2}
89 1+864.T+7.04e5T2 1 + 864.T + 7.04e5T^{2}
97 1+614.T+9.12e5T2 1 + 614.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

−8.986622244657292978343557554888, −8.090174790929738396339932187172, −7.50794226420143644569768480356, −6.67740543191907973307549064264, −5.56339976854934418787985284595, −5.01638243829296425976353449211, −2.90525341950227781367710072641, −2.27778088135269098477359623677, −1.22537985344270257706748135095, 0, 1.22537985344270257706748135095, 2.27778088135269098477359623677, 2.90525341950227781367710072641, 5.01638243829296425976353449211, 5.56339976854934418787985284595, 6.67740543191907973307549064264, 7.50794226420143644569768480356, 8.090174790929738396339932187172, 8.986622244657292978343557554888

Graph of the ZZ-function along the critical line