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Properties

Label 2-1323-1.1-c3-0-70
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  − 3.68·2-s + 5.59·4-s − 11.8·5-s + 8.85·8-s + 43.7·10-s − 22.2·11-s − 75.1·13-s − 77.4·16-s + 74.1·17-s + 7.41·19-s − 66.4·20-s + 81.9·22-s + 205.·23-s + 15.9·25-s + 276.·26-s − 148.·29-s + 164.·31-s + 214.·32-s − 273.·34-s − 205.·37-s − 27.3·38-s − 105.·40-s + 83.9·41-s − 368.·43-s − 124.·44-s − 758.·46-s + 98.3·47-s + ⋯
L(s)  = 1  − 1.30·2-s + 0.699·4-s − 1.06·5-s + 0.391·8-s + 1.38·10-s − 0.608·11-s − 1.60·13-s − 1.21·16-s + 1.05·17-s + 0.0895·19-s − 0.743·20-s + 0.793·22-s + 1.86·23-s + 0.127·25-s + 2.08·26-s − 0.949·29-s + 0.952·31-s + 1.18·32-s − 1.37·34-s − 0.911·37-s − 0.116·38-s − 0.415·40-s + 0.319·41-s − 1.30·43-s − 0.426·44-s − 2.43·46-s + 0.305·47-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+3.68T+8T2 1 + 3.68T + 8T^{2}
5 1+11.8T+125T2 1 + 11.8T + 125T^{2}
11 1+22.2T+1.33e3T2 1 + 22.2T + 1.33e3T^{2}
13 1+75.1T+2.19e3T2 1 + 75.1T + 2.19e3T^{2}
17 1−74.1T+4.91e3T2 1 - 74.1T + 4.91e3T^{2}
19 1−7.41T+6.85e3T2 1 - 7.41T + 6.85e3T^{2}
23 1−205.T+1.21e4T2 1 - 205.T + 1.21e4T^{2}
29 1+148.T+2.43e4T2 1 + 148.T + 2.43e4T^{2}
31 1−164.T+2.97e4T2 1 - 164.T + 2.97e4T^{2}
37 1+205.T+5.06e4T2 1 + 205.T + 5.06e4T^{2}
41 1−83.9T+6.89e4T2 1 - 83.9T + 6.89e4T^{2}
43 1+368.T+7.95e4T2 1 + 368.T + 7.95e4T^{2}
47 1−98.3T+1.03e5T2 1 - 98.3T + 1.03e5T^{2}
53 1−293.T+1.48e5T2 1 - 293.T + 1.48e5T^{2}
59 1−509.T+2.05e5T2 1 - 509.T + 2.05e5T^{2}
61 1−696.T+2.26e5T2 1 - 696.T + 2.26e5T^{2}
67 1−370.T+3.00e5T2 1 - 370.T + 3.00e5T^{2}
71 1+121.T+3.57e5T2 1 + 121.T + 3.57e5T^{2}
73 1−682.T+3.89e5T2 1 - 682.T + 3.89e5T^{2}
79 1+669.T+4.93e5T2 1 + 669.T + 4.93e5T^{2}
83 1+1.18e3T+5.71e5T2 1 + 1.18e3T + 5.71e5T^{2}
89 1+598.T+7.04e5T2 1 + 598.T + 7.04e5T^{2}
97 1−1.03e3T+9.12e5T2 1 - 1.03e3T + 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

−8.716825033313170160833453516212, −8.147539251295607173850099776354, −7.22934923458286036095388404925, −7.15856371889363806204943324487, −5.36298718312043907799187142132, −4.66973403249598545827425700627, −3.44245629685745685875045094732, −2.33702165415097516486525467925, −0.910064134066559161033882164644, 0, 0.910064134066559161033882164644, 2.33702165415097516486525467925, 3.44245629685745685875045094732, 4.66973403249598545827425700627, 5.36298718312043907799187142132, 7.15856371889363806204943324487, 7.22934923458286036095388404925, 8.147539251295607173850099776354, 8.716825033313170160833453516212

Graph of the ZZ-function along the critical line