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Properties

Label 2-1323-1.1-c3-0-69
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  − 2.49·2-s − 1.76·4-s + 14.5·5-s + 24.3·8-s − 36.4·10-s − 0.0794·11-s + 86.9·13-s − 46.7·16-s + 93.0·17-s − 147.·19-s − 25.7·20-s + 0.198·22-s + 154.·23-s + 87.7·25-s − 217.·26-s + 205.·29-s + 271.·31-s − 78.3·32-s − 232.·34-s + 48.3·37-s + 369.·38-s + 355.·40-s + 52.3·41-s + 48.0·43-s + 0.140·44-s − 385.·46-s − 266.·47-s + ⋯
L(s)  = 1  − 0.882·2-s − 0.220·4-s + 1.30·5-s + 1.07·8-s − 1.15·10-s − 0.00217·11-s + 1.85·13-s − 0.730·16-s + 1.32·17-s − 1.78·19-s − 0.288·20-s + 0.00192·22-s + 1.39·23-s + 0.701·25-s − 1.63·26-s + 1.31·29-s + 1.57·31-s − 0.433·32-s − 1.17·34-s + 0.214·37-s + 1.57·38-s + 1.40·40-s + 0.199·41-s + 0.170·43-s + 0.000480·44-s − 1.23·46-s − 0.825·47-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 1.9355617801.935561780
L(12)L(\frac12) ≈\approx 1.9355617801.935561780
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+2.49T+8T2 1 + 2.49T + 8T^{2}
5 1−14.5T+125T2 1 - 14.5T + 125T^{2}
11 1+0.0794T+1.33e3T2 1 + 0.0794T + 1.33e3T^{2}
13 1−86.9T+2.19e3T2 1 - 86.9T + 2.19e3T^{2}
17 1−93.0T+4.91e3T2 1 - 93.0T + 4.91e3T^{2}
19 1+147.T+6.85e3T2 1 + 147.T + 6.85e3T^{2}
23 1−154.T+1.21e4T2 1 - 154.T + 1.21e4T^{2}
29 1−205.T+2.43e4T2 1 - 205.T + 2.43e4T^{2}
31 1−271.T+2.97e4T2 1 - 271.T + 2.97e4T^{2}
37 1−48.3T+5.06e4T2 1 - 48.3T + 5.06e4T^{2}
41 1−52.3T+6.89e4T2 1 - 52.3T + 6.89e4T^{2}
43 1−48.0T+7.95e4T2 1 - 48.0T + 7.95e4T^{2}
47 1+266.T+1.03e5T2 1 + 266.T + 1.03e5T^{2}
53 1−11.8T+1.48e5T2 1 - 11.8T + 1.48e5T^{2}
59 1+542.T+2.05e5T2 1 + 542.T + 2.05e5T^{2}
61 1−84.4T+2.26e5T2 1 - 84.4T + 2.26e5T^{2}
67 1+51.0T+3.00e5T2 1 + 51.0T + 3.00e5T^{2}
71 1−495.T+3.57e5T2 1 - 495.T + 3.57e5T^{2}
73 1−71.1T+3.89e5T2 1 - 71.1T + 3.89e5T^{2}
79 1−378.T+4.93e5T2 1 - 378.T + 4.93e5T^{2}
83 1+486.T+5.71e5T2 1 + 486.T + 5.71e5T^{2}
89 1+1.16e3T+7.04e5T2 1 + 1.16e3T + 7.04e5T^{2}
97 1−1.31e3T+9.12e5T2 1 - 1.31e3T + 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

−9.232604390673024542310175417159, −8.527137108928056810930408162870, −8.057755545431497260423609901463, −6.66471062806954564738825877904, −6.14197633716183375129192531482, −5.14052044914316208690829376696, −4.15430607284986968739286354749, −2.86773863087654842400470117102, −1.55240959726617889010219230939, −0.907769866796201208928267673463, 0.907769866796201208928267673463, 1.55240959726617889010219230939, 2.86773863087654842400470117102, 4.15430607284986968739286354749, 5.14052044914316208690829376696, 6.14197633716183375129192531482, 6.66471062806954564738825877904, 8.057755545431497260423609901463, 8.527137108928056810930408162870, 9.232604390673024542310175417159

Graph of the ZZ-function along the critical line