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Properties

Label 2-1323-1.1-c1-0-17
Degree 22
Conductor 13231323
Sign 11
Analytic cond. 10.564210.5642
Root an. cond. 3.250263.25026
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 3·5-s + 3·8-s − 3·10-s + 5·11-s + 6·13-s − 16-s − 6·17-s + 3·19-s − 3·20-s − 5·22-s + 23-s + 4·25-s − 6·26-s − 2·29-s − 3·31-s − 5·32-s + 6·34-s + 3·37-s − 3·38-s + 9·40-s − 9·41-s + 6·43-s − 5·44-s − 46-s + 6·47-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.34·5-s + 1.06·8-s − 0.948·10-s + 1.50·11-s + 1.66·13-s − 1/4·16-s − 1.45·17-s + 0.688·19-s − 0.670·20-s − 1.06·22-s + 0.208·23-s + 4/5·25-s − 1.17·26-s − 0.371·29-s − 0.538·31-s − 0.883·32-s + 1.02·34-s + 0.493·37-s − 0.486·38-s + 1.42·40-s − 1.40·41-s + 0.914·43-s − 0.753·44-s − 0.147·46-s + 0.875·47-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 11
Analytic conductor: 10.564210.5642
Root analytic conductor: 3.250263.25026
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1323, ( :1/2), 1)(2,\ 1323,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4912932981.491293298
L(12)L(\frac12) ≈\approx 1.4912932981.491293298
L(32)L(\frac{3}{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+T+pT2 1 + T + p T^{2}
5 1−3T+pT2 1 - 3 T + p T^{2}
11 1−5T+pT2 1 - 5 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−3T+pT2 1 - 3 T + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1−3T+pT2 1 - 3 T + p T^{2}
41 1+9T+pT2 1 + 9 T + p T^{2}
43 1−6T+pT2 1 - 6 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1+8T+pT2 1 + 8 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1−12T+pT2 1 - 12 T + p T^{2}
67 1+14T+pT2 1 + 14 T + p T^{2}
71 1+7T+pT2 1 + 7 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1−3T+pT2 1 - 3 T + p T^{2}
97 1−12T+pT2 1 - 12 T + p T^{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

−9.309598966834482564926917981283, −9.077294699564776327557372455758, −8.413589450998261439447207221768, −7.13167564560012152294416600100, −6.34215171511087571525667957813, −5.63752173484997695290532171122, −4.47191624657260288271626711721, −3.59632548449757607506915649618, −1.92610841189085090066739675799, −1.12292208190814267208777122415, 1.12292208190814267208777122415, 1.92610841189085090066739675799, 3.59632548449757607506915649618, 4.47191624657260288271626711721, 5.63752173484997695290532171122, 6.34215171511087571525667957813, 7.13167564560012152294416600100, 8.413589450998261439447207221768, 9.077294699564776327557372455758, 9.309598966834482564926917981283

Graph of the ZZ-function along the critical line