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Properties

Label 2-1323-1.1-c3-0-106
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  − 4.70·2-s + 14.1·4-s − 0.830·5-s − 29.0·8-s + 3.90·10-s + 68.2·11-s − 35.7·13-s + 23.2·16-s − 14.6·17-s + 86.7·19-s − 11.7·20-s − 321.·22-s − 58.4·23-s − 124.·25-s + 168.·26-s + 67.1·29-s − 69.5·31-s + 122.·32-s + 69.0·34-s − 289.·37-s − 408.·38-s + 24.0·40-s + 357.·41-s − 235.·43-s + 966.·44-s + 274.·46-s − 450.·47-s + ⋯
L(s)  = 1  − 1.66·2-s + 1.77·4-s − 0.0742·5-s − 1.28·8-s + 0.123·10-s + 1.87·11-s − 0.763·13-s + 0.363·16-s − 0.209·17-s + 1.04·19-s − 0.131·20-s − 3.11·22-s − 0.529·23-s − 0.994·25-s + 1.27·26-s + 0.429·29-s − 0.402·31-s + 0.677·32-s + 0.348·34-s − 1.28·37-s − 1.74·38-s + 0.0952·40-s + 1.36·41-s − 0.836·43-s + 3.31·44-s + 0.881·46-s − 1.39·47-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+4.70T+8T2 1 + 4.70T + 8T^{2}
5 1+0.830T+125T2 1 + 0.830T + 125T^{2}
11 1−68.2T+1.33e3T2 1 - 68.2T + 1.33e3T^{2}
13 1+35.7T+2.19e3T2 1 + 35.7T + 2.19e3T^{2}
17 1+14.6T+4.91e3T2 1 + 14.6T + 4.91e3T^{2}
19 1−86.7T+6.85e3T2 1 - 86.7T + 6.85e3T^{2}
23 1+58.4T+1.21e4T2 1 + 58.4T + 1.21e4T^{2}
29 1−67.1T+2.43e4T2 1 - 67.1T + 2.43e4T^{2}
31 1+69.5T+2.97e4T2 1 + 69.5T + 2.97e4T^{2}
37 1+289.T+5.06e4T2 1 + 289.T + 5.06e4T^{2}
41 1−357.T+6.89e4T2 1 - 357.T + 6.89e4T^{2}
43 1+235.T+7.95e4T2 1 + 235.T + 7.95e4T^{2}
47 1+450.T+1.03e5T2 1 + 450.T + 1.03e5T^{2}
53 1+172.T+1.48e5T2 1 + 172.T + 1.48e5T^{2}
59 1−445.T+2.05e5T2 1 - 445.T + 2.05e5T^{2}
61 1−476.T+2.26e5T2 1 - 476.T + 2.26e5T^{2}
67 1+804.T+3.00e5T2 1 + 804.T + 3.00e5T^{2}
71 1+353.T+3.57e5T2 1 + 353.T + 3.57e5T^{2}
73 1+778.T+3.89e5T2 1 + 778.T + 3.89e5T^{2}
79 1+74.1T+4.93e5T2 1 + 74.1T + 4.93e5T^{2}
83 1−1.16e3T+5.71e5T2 1 - 1.16e3T + 5.71e5T^{2}
89 1+1.51e3T+7.04e5T2 1 + 1.51e3T + 7.04e5T^{2}
97 1−1.63e3T+9.12e5T2 1 - 1.63e3T + 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

−8.962134227187167938587944165383, −8.217924648099309367778758767659, −7.32786349813478456856363904421, −6.79593053980232161323010570096, −5.86396002869175492884116174523, −4.49871053815137030683181201711, −3.38065834222808802547182986241, −2.01668180266059204247380194139, −1.19670716109598163394732490012, 0, 1.19670716109598163394732490012, 2.01668180266059204247380194139, 3.38065834222808802547182986241, 4.49871053815137030683181201711, 5.86396002869175492884116174523, 6.79593053980232161323010570096, 7.32786349813478456856363904421, 8.217924648099309367778758767659, 8.962134227187167938587944165383

Graph of the ZZ-function along the critical line