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Properties

Label 2-1323-1.1-c3-0-36
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  − 4.73·2-s + 14.3·4-s + 9.92·5-s − 30.2·8-s − 46.9·10-s + 3.71·11-s + 15.5·13-s + 28.0·16-s − 33.4·17-s − 135.·19-s + 142.·20-s − 17.5·22-s + 87.7·23-s − 26.4·25-s − 73.6·26-s − 242.·29-s + 194.·31-s + 109.·32-s + 158.·34-s − 239.·37-s + 643.·38-s − 300.·40-s + 470.·41-s − 448.·43-s + 53.4·44-s − 415.·46-s + 4.15·47-s + ⋯
L(s)  = 1  − 1.67·2-s + 1.79·4-s + 0.888·5-s − 1.33·8-s − 1.48·10-s + 0.101·11-s + 0.332·13-s + 0.437·16-s − 0.477·17-s − 1.64·19-s + 1.59·20-s − 0.170·22-s + 0.795·23-s − 0.211·25-s − 0.555·26-s − 1.55·29-s + 1.12·31-s + 0.604·32-s + 0.799·34-s − 1.06·37-s + 2.74·38-s − 1.18·40-s + 1.79·41-s − 1.58·43-s + 0.183·44-s − 1.33·46-s + 0.0129·47-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 0.92796320440.9279632044
L(12)L(\frac12) ≈\approx 0.92796320440.9279632044
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+4.73T+8T2 1 + 4.73T + 8T^{2}
5 1−9.92T+125T2 1 - 9.92T + 125T^{2}
11 1−3.71T+1.33e3T2 1 - 3.71T + 1.33e3T^{2}
13 1−15.5T+2.19e3T2 1 - 15.5T + 2.19e3T^{2}
17 1+33.4T+4.91e3T2 1 + 33.4T + 4.91e3T^{2}
19 1+135.T+6.85e3T2 1 + 135.T + 6.85e3T^{2}
23 1−87.7T+1.21e4T2 1 - 87.7T + 1.21e4T^{2}
29 1+242.T+2.43e4T2 1 + 242.T + 2.43e4T^{2}
31 1−194.T+2.97e4T2 1 - 194.T + 2.97e4T^{2}
37 1+239.T+5.06e4T2 1 + 239.T + 5.06e4T^{2}
41 1−470.T+6.89e4T2 1 - 470.T + 6.89e4T^{2}
43 1+448.T+7.95e4T2 1 + 448.T + 7.95e4T^{2}
47 1−4.15T+1.03e5T2 1 - 4.15T + 1.03e5T^{2}
53 1−736.T+1.48e5T2 1 - 736.T + 1.48e5T^{2}
59 1−279.T+2.05e5T2 1 - 279.T + 2.05e5T^{2}
61 1−514.T+2.26e5T2 1 - 514.T + 2.26e5T^{2}
67 1+102.T+3.00e5T2 1 + 102.T + 3.00e5T^{2}
71 1−44.1T+3.57e5T2 1 - 44.1T + 3.57e5T^{2}
73 1−901.T+3.89e5T2 1 - 901.T + 3.89e5T^{2}
79 1−1.05e3T+4.93e5T2 1 - 1.05e3T + 4.93e5T^{2}
83 1−487.T+5.71e5T2 1 - 487.T + 5.71e5T^{2}
89 1−963.T+7.04e5T2 1 - 963.T + 7.04e5T^{2}
97 1+726.T+9.12e5T2 1 + 726.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

−9.201469306942394824899384266169, −8.656354916581939464317960395147, −7.908415229351685857426441268681, −6.85153724283759710360390840084, −6.37468083428123473232078564898, −5.31722524763747337099803341728, −3.97084032343759934370877409845, −2.42981308880365942153631857327, −1.81117132762125842445054730865, −0.61833833769799951755914455817, 0.61833833769799951755914455817, 1.81117132762125842445054730865, 2.42981308880365942153631857327, 3.97084032343759934370877409845, 5.31722524763747337099803341728, 6.37468083428123473232078564898, 6.85153724283759710360390840084, 7.908415229351685857426441268681, 8.656354916581939464317960395147, 9.201469306942394824899384266169

Graph of the ZZ-function along the critical line