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Properties

Label 2-1323-1.1-c3-0-56
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.62·2-s + 5.17·4-s + 4.84·5-s + 10.2·8-s − 17.5·10-s − 20.1·11-s + 72.2·13-s − 78.6·16-s + 132.·17-s + 76.9·19-s + 25.0·20-s + 73.0·22-s − 22.4·23-s − 101.·25-s − 262.·26-s + 193.·29-s + 89.7·31-s + 203.·32-s − 480.·34-s + 47.9·37-s − 279.·38-s + 49.6·40-s + 3.41·41-s − 168.·43-s − 104.·44-s + 81.5·46-s + 163.·47-s + ⋯
L(s)  = 1  − 1.28·2-s + 0.646·4-s + 0.433·5-s + 0.453·8-s − 0.555·10-s − 0.551·11-s + 1.54·13-s − 1.22·16-s + 1.88·17-s + 0.928·19-s + 0.280·20-s + 0.708·22-s − 0.203·23-s − 0.812·25-s − 1.97·26-s + 1.23·29-s + 0.520·31-s + 1.12·32-s − 2.42·34-s + 0.212·37-s − 1.19·38-s + 0.196·40-s + 0.0130·41-s − 0.597·43-s − 0.356·44-s + 0.261·46-s + 0.506·47-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 1.3692415401.369241540
L(12)L(\frac12) ≈\approx 1.3692415401.369241540
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+3.62T+8T2 1 + 3.62T + 8T^{2}
5 1−4.84T+125T2 1 - 4.84T + 125T^{2}
11 1+20.1T+1.33e3T2 1 + 20.1T + 1.33e3T^{2}
13 1−72.2T+2.19e3T2 1 - 72.2T + 2.19e3T^{2}
17 1−132.T+4.91e3T2 1 - 132.T + 4.91e3T^{2}
19 1−76.9T+6.85e3T2 1 - 76.9T + 6.85e3T^{2}
23 1+22.4T+1.21e4T2 1 + 22.4T + 1.21e4T^{2}
29 1−193.T+2.43e4T2 1 - 193.T + 2.43e4T^{2}
31 1−89.7T+2.97e4T2 1 - 89.7T + 2.97e4T^{2}
37 1−47.9T+5.06e4T2 1 - 47.9T + 5.06e4T^{2}
41 1−3.41T+6.89e4T2 1 - 3.41T + 6.89e4T^{2}
43 1+168.T+7.95e4T2 1 + 168.T + 7.95e4T^{2}
47 1−163.T+1.03e5T2 1 - 163.T + 1.03e5T^{2}
53 1−337.T+1.48e5T2 1 - 337.T + 1.48e5T^{2}
59 1−517.T+2.05e5T2 1 - 517.T + 2.05e5T^{2}
61 1−424.T+2.26e5T2 1 - 424.T + 2.26e5T^{2}
67 1+978.T+3.00e5T2 1 + 978.T + 3.00e5T^{2}
71 1+40.4T+3.57e5T2 1 + 40.4T + 3.57e5T^{2}
73 1−482.T+3.89e5T2 1 - 482.T + 3.89e5T^{2}
79 1+1.07e3T+4.93e5T2 1 + 1.07e3T + 4.93e5T^{2}
83 1+811.T+5.71e5T2 1 + 811.T + 5.71e5T^{2}
89 1−997.T+7.04e5T2 1 - 997.T + 7.04e5T^{2}
97 1+1.45e3T+9.12e5T2 1 + 1.45e3T + 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

−9.290549771325094497698059541275, −8.389277054240070183558111763720, −7.962356323089872201793313561246, −7.08466907486733435823596208125, −6.00685415061052921525557230452, −5.29105513808097321887272781681, −3.97215174945320474629585742124, −2.85947737409268625486610409180, −1.49022794145752444118028556562, −0.801622842332916639903454678476, 0.801622842332916639903454678476, 1.49022794145752444118028556562, 2.85947737409268625486610409180, 3.97215174945320474629585742124, 5.29105513808097321887272781681, 6.00685415061052921525557230452, 7.08466907486733435823596208125, 7.962356323089872201793313561246, 8.389277054240070183558111763720, 9.290549771325094497698059541275

Graph of the ZZ-function along the critical line