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Properties

Label 2-1323-1.1-c3-0-140
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  + 1.63·2-s − 5.33·4-s + 12.3·5-s − 21.7·8-s + 20.2·10-s + 29.0·11-s + 52.9·13-s + 7.18·16-s − 122.·17-s − 141.·19-s − 66.0·20-s + 47.3·22-s − 60.2·23-s + 28.2·25-s + 86.4·26-s + 126.·29-s − 150.·31-s + 185.·32-s − 199.·34-s − 341.·37-s − 230.·38-s − 269.·40-s + 292.·41-s + 290.·43-s − 154.·44-s − 98.3·46-s + 284.·47-s + ⋯
L(s)  = 1  + 0.576·2-s − 0.667·4-s + 1.10·5-s − 0.961·8-s + 0.638·10-s + 0.795·11-s + 1.13·13-s + 0.112·16-s − 1.74·17-s − 1.70·19-s − 0.738·20-s + 0.458·22-s − 0.546·23-s + 0.225·25-s + 0.652·26-s + 0.813·29-s − 0.873·31-s + 1.02·32-s − 1.00·34-s − 1.51·37-s − 0.982·38-s − 1.06·40-s + 1.11·41-s + 1.03·43-s − 0.530·44-s − 0.315·46-s + 0.881·47-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 1−1.63T+8T2 1 - 1.63T + 8T^{2}
5 1−12.3T+125T2 1 - 12.3T + 125T^{2}
11 1−29.0T+1.33e3T2 1 - 29.0T + 1.33e3T^{2}
13 1−52.9T+2.19e3T2 1 - 52.9T + 2.19e3T^{2}
17 1+122.T+4.91e3T2 1 + 122.T + 4.91e3T^{2}
19 1+141.T+6.85e3T2 1 + 141.T + 6.85e3T^{2}
23 1+60.2T+1.21e4T2 1 + 60.2T + 1.21e4T^{2}
29 1−126.T+2.43e4T2 1 - 126.T + 2.43e4T^{2}
31 1+150.T+2.97e4T2 1 + 150.T + 2.97e4T^{2}
37 1+341.T+5.06e4T2 1 + 341.T + 5.06e4T^{2}
41 1−292.T+6.89e4T2 1 - 292.T + 6.89e4T^{2}
43 1−290.T+7.95e4T2 1 - 290.T + 7.95e4T^{2}
47 1−284.T+1.03e5T2 1 - 284.T + 1.03e5T^{2}
53 1+387.T+1.48e5T2 1 + 387.T + 1.48e5T^{2}
59 1−269.T+2.05e5T2 1 - 269.T + 2.05e5T^{2}
61 1−239.T+2.26e5T2 1 - 239.T + 2.26e5T^{2}
67 1+712.T+3.00e5T2 1 + 712.T + 3.00e5T^{2}
71 1+270.T+3.57e5T2 1 + 270.T + 3.57e5T^{2}
73 1+146.T+3.89e5T2 1 + 146.T + 3.89e5T^{2}
79 1+652.T+4.93e5T2 1 + 652.T + 4.93e5T^{2}
83 1+35.0T+5.71e5T2 1 + 35.0T + 5.71e5T^{2}
89 1+1.39e3T+7.04e5T2 1 + 1.39e3T + 7.04e5T^{2}
97 1+805.T+9.12e5T2 1 + 805.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

−8.874171026592983364970462415918, −8.438165542169460199597778921662, −6.81639981116577737478348706469, −6.18467686629707301649257214343, −5.62509665454960362738627244671, −4.36961827561430713668416298739, −3.96854712322408522064097291236, −2.54904261613481540568649223430, −1.53718974564308458735721115539, 0, 1.53718974564308458735721115539, 2.54904261613481540568649223430, 3.96854712322408522064097291236, 4.36961827561430713668416298739, 5.62509665454960362738627244671, 6.18467686629707301649257214343, 6.81639981116577737478348706469, 8.438165542169460199597778921662, 8.874171026592983364970462415918

Graph of the ZZ-function along the critical line