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Properties

Label 2-1323-1.1-c3-0-23
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  + 1.55·2-s − 5.59·4-s − 9.81·5-s − 21.0·8-s − 15.2·10-s + 70.6·11-s − 55.5·13-s + 12.0·16-s − 13.4·17-s − 91.3·19-s + 54.8·20-s + 109.·22-s − 113.·23-s − 28.7·25-s − 86.1·26-s − 12.4·29-s − 222.·31-s + 187.·32-s − 20.8·34-s + 257.·37-s − 141.·38-s + 206.·40-s − 286.·41-s + 4.81·43-s − 394.·44-s − 176.·46-s + 609.·47-s + ⋯
L(s)  = 1  + 0.548·2-s − 0.699·4-s − 0.877·5-s − 0.931·8-s − 0.481·10-s + 1.93·11-s − 1.18·13-s + 0.188·16-s − 0.191·17-s − 1.10·19-s + 0.613·20-s + 1.06·22-s − 1.03·23-s − 0.229·25-s − 0.650·26-s − 0.0800·29-s − 1.29·31-s + 1.03·32-s − 0.104·34-s + 1.14·37-s − 0.605·38-s + 0.817·40-s − 1.09·41-s + 0.0170·43-s − 1.35·44-s − 0.565·46-s + 1.89·47-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 1.1824768881.182476888
L(12)L(\frac12) ≈\approx 1.1824768881.182476888
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 1−1.55T+8T2 1 - 1.55T + 8T^{2}
5 1+9.81T+125T2 1 + 9.81T + 125T^{2}
11 1−70.6T+1.33e3T2 1 - 70.6T + 1.33e3T^{2}
13 1+55.5T+2.19e3T2 1 + 55.5T + 2.19e3T^{2}
17 1+13.4T+4.91e3T2 1 + 13.4T + 4.91e3T^{2}
19 1+91.3T+6.85e3T2 1 + 91.3T + 6.85e3T^{2}
23 1+113.T+1.21e4T2 1 + 113.T + 1.21e4T^{2}
29 1+12.4T+2.43e4T2 1 + 12.4T + 2.43e4T^{2}
31 1+222.T+2.97e4T2 1 + 222.T + 2.97e4T^{2}
37 1−257.T+5.06e4T2 1 - 257.T + 5.06e4T^{2}
41 1+286.T+6.89e4T2 1 + 286.T + 6.89e4T^{2}
43 1−4.81T+7.95e4T2 1 - 4.81T + 7.95e4T^{2}
47 1−609.T+1.03e5T2 1 - 609.T + 1.03e5T^{2}
53 1+691.T+1.48e5T2 1 + 691.T + 1.48e5T^{2}
59 1+217.T+2.05e5T2 1 + 217.T + 2.05e5T^{2}
61 1−764.T+2.26e5T2 1 - 764.T + 2.26e5T^{2}
67 1+98.5T+3.00e5T2 1 + 98.5T + 3.00e5T^{2}
71 1−921.T+3.57e5T2 1 - 921.T + 3.57e5T^{2}
73 1−219.T+3.89e5T2 1 - 219.T + 3.89e5T^{2}
79 1+9.42T+4.93e5T2 1 + 9.42T + 4.93e5T^{2}
83 1−800.T+5.71e5T2 1 - 800.T + 5.71e5T^{2}
89 1−253.T+7.04e5T2 1 - 253.T + 7.04e5T^{2}
97 1−59.2T+9.12e5T2 1 - 59.2T + 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

−9.268491110835052559365642046344, −8.504644085997278823099769013714, −7.65539286313107260325773739977, −6.69081232173316504816994556883, −5.91351902338233832174905264054, −4.74994257986462456538914680011, −4.07000984457363537420500899047, −3.57523787172336859884453511744, −2.06755728409460973295178542563, −0.49306999778494320234740779790, 0.49306999778494320234740779790, 2.06755728409460973295178542563, 3.57523787172336859884453511744, 4.07000984457363537420500899047, 4.74994257986462456538914680011, 5.91351902338233832174905264054, 6.69081232173316504816994556883, 7.65539286313107260325773739977, 8.504644085997278823099769013714, 9.268491110835052559365642046344

Graph of the ZZ-function along the critical line