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Properties

Label 2-1323-1.1-c3-0-150
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  + 4.49·2-s + 12.1·4-s − 8.19·5-s + 18.8·8-s − 36.8·10-s − 33.3·11-s + 54.5·13-s − 13.0·16-s − 1.19·17-s + 124.·19-s − 99.8·20-s − 150.·22-s − 133.·23-s − 57.7·25-s + 245.·26-s − 145.·29-s − 78.1·31-s − 208.·32-s − 5.38·34-s − 134.·37-s + 558.·38-s − 154.·40-s − 178.·41-s + 211.·43-s − 406.·44-s − 602.·46-s − 518.·47-s + ⋯
L(s)  = 1  + 1.58·2-s + 1.52·4-s − 0.733·5-s + 0.830·8-s − 1.16·10-s − 0.915·11-s + 1.16·13-s − 0.203·16-s − 0.0171·17-s + 1.50·19-s − 1.11·20-s − 1.45·22-s − 1.21·23-s − 0.462·25-s + 1.85·26-s − 0.932·29-s − 0.452·31-s − 1.15·32-s − 0.0271·34-s − 0.598·37-s + 2.38·38-s − 0.609·40-s − 0.679·41-s + 0.748·43-s − 1.39·44-s − 1.92·46-s − 1.60·47-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 1−4.49T+8T2 1 - 4.49T + 8T^{2}
5 1+8.19T+125T2 1 + 8.19T + 125T^{2}
11 1+33.3T+1.33e3T2 1 + 33.3T + 1.33e3T^{2}
13 1−54.5T+2.19e3T2 1 - 54.5T + 2.19e3T^{2}
17 1+1.19T+4.91e3T2 1 + 1.19T + 4.91e3T^{2}
19 1−124.T+6.85e3T2 1 - 124.T + 6.85e3T^{2}
23 1+133.T+1.21e4T2 1 + 133.T + 1.21e4T^{2}
29 1+145.T+2.43e4T2 1 + 145.T + 2.43e4T^{2}
31 1+78.1T+2.97e4T2 1 + 78.1T + 2.97e4T^{2}
37 1+134.T+5.06e4T2 1 + 134.T + 5.06e4T^{2}
41 1+178.T+6.89e4T2 1 + 178.T + 6.89e4T^{2}
43 1−211.T+7.95e4T2 1 - 211.T + 7.95e4T^{2}
47 1+518.T+1.03e5T2 1 + 518.T + 1.03e5T^{2}
53 1−269.T+1.48e5T2 1 - 269.T + 1.48e5T^{2}
59 1+306.T+2.05e5T2 1 + 306.T + 2.05e5T^{2}
61 1+38.0T+2.26e5T2 1 + 38.0T + 2.26e5T^{2}
67 1−610.T+3.00e5T2 1 - 610.T + 3.00e5T^{2}
71 1+1.08e3T+3.57e5T2 1 + 1.08e3T + 3.57e5T^{2}
73 1+1.08e3T+3.89e5T2 1 + 1.08e3T + 3.89e5T^{2}
79 1+1.19e3T+4.93e5T2 1 + 1.19e3T + 4.93e5T^{2}
83 1+150.T+5.71e5T2 1 + 150.T + 5.71e5T^{2}
89 1−221.T+7.04e5T2 1 - 221.T + 7.04e5T^{2}
97 1−1.60e3T+9.12e5T2 1 - 1.60e3T + 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

−8.680305850150465548539146802624, −7.77505762193283109661809868763, −7.12339391432473026597989086552, −5.94942413269228364629466529952, −5.50249393359753868369871466895, −4.48319752255775413531902372464, −3.66698260874981768601655991106, −3.07726222332152190730412189863, −1.75116663222038568429291041664, 0, 1.75116663222038568429291041664, 3.07726222332152190730412189863, 3.66698260874981768601655991106, 4.48319752255775413531902372464, 5.50249393359753868369871466895, 5.94942413269228364629466529952, 7.12339391432473026597989086552, 7.77505762193283109661809868763, 8.680305850150465548539146802624

Graph of the ZZ-function along the critical line