↑

Properties

Label 2-1323-1.1-c3-0-11
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.49·2-s + 4.19·4-s + 7.87·5-s + 13.2·8-s − 27.5·10-s − 54.0·11-s − 48.9·13-s − 79.9·16-s − 96.9·17-s − 142.·19-s + 33.0·20-s + 188.·22-s − 103.·23-s − 62.9·25-s + 170.·26-s − 24.5·29-s + 187.·31-s + 172.·32-s + 338.·34-s + 146.·37-s + 497.·38-s + 104.·40-s − 314.·41-s + 173.·43-s − 226.·44-s + 362.·46-s + 259.·47-s + ⋯
L(s)  = 1  − 1.23·2-s + 0.524·4-s + 0.704·5-s + 0.587·8-s − 0.869·10-s − 1.48·11-s − 1.04·13-s − 1.24·16-s − 1.38·17-s − 1.71·19-s + 0.369·20-s + 1.82·22-s − 0.940·23-s − 0.503·25-s + 1.28·26-s − 0.157·29-s + 1.08·31-s + 0.955·32-s + 1.70·34-s + 0.651·37-s + 2.12·38-s + 0.413·40-s − 1.19·41-s + 0.614·43-s − 0.776·44-s + 1.16·46-s + 0.804·47-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 0.35628445000.3562844500
L(12)L(\frac12) ≈\approx 0.35628445000.3562844500
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+3.49T+8T2 1 + 3.49T + 8T^{2}
5 1−7.87T+125T2 1 - 7.87T + 125T^{2}
11 1+54.0T+1.33e3T2 1 + 54.0T + 1.33e3T^{2}
13 1+48.9T+2.19e3T2 1 + 48.9T + 2.19e3T^{2}
17 1+96.9T+4.91e3T2 1 + 96.9T + 4.91e3T^{2}
19 1+142.T+6.85e3T2 1 + 142.T + 6.85e3T^{2}
23 1+103.T+1.21e4T2 1 + 103.T + 1.21e4T^{2}
29 1+24.5T+2.43e4T2 1 + 24.5T + 2.43e4T^{2}
31 1−187.T+2.97e4T2 1 - 187.T + 2.97e4T^{2}
37 1−146.T+5.06e4T2 1 - 146.T + 5.06e4T^{2}
41 1+314.T+6.89e4T2 1 + 314.T + 6.89e4T^{2}
43 1−173.T+7.95e4T2 1 - 173.T + 7.95e4T^{2}
47 1−259.T+1.03e5T2 1 - 259.T + 1.03e5T^{2}
53 1+620.T+1.48e5T2 1 + 620.T + 1.48e5T^{2}
59 1−443.T+2.05e5T2 1 - 443.T + 2.05e5T^{2}
61 1−113.T+2.26e5T2 1 - 113.T + 2.26e5T^{2}
67 1−628.T+3.00e5T2 1 - 628.T + 3.00e5T^{2}
71 1+41.3T+3.57e5T2 1 + 41.3T + 3.57e5T^{2}
73 1−447.T+3.89e5T2 1 - 447.T + 3.89e5T^{2}
79 1−434.T+4.93e5T2 1 - 434.T + 4.93e5T^{2}
83 1−329.T+5.71e5T2 1 - 329.T + 5.71e5T^{2}
89 1−24.8T+7.04e5T2 1 - 24.8T + 7.04e5T^{2}
97 1+499.T+9.12e5T2 1 + 499.T + 9.12e5T^{2}
show more
show less
   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

−9.372945560846556579223907623160, −8.374769461188760483046962842362, −7.990189338735132467873071635436, −6.97543623254963569369277121786, −6.16797279585251078814860166172, −5.02427307010313947474787082723, −4.28088690482914133229388945107, −2.38256583256639514564599558328, −2.07839319178881540484981670390, −0.33833020353531410404616194610, 0.33833020353531410404616194610, 2.07839319178881540484981670390, 2.38256583256639514564599558328, 4.28088690482914133229388945107, 5.02427307010313947474787082723, 6.16797279585251078814860166172, 6.97543623254963569369277121786, 7.990189338735132467873071635436, 8.374769461188760483046962842362, 9.372945560846556579223907623160

Graph of the ZZ-function along the critical line