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Properties

Label 2-1323-1.1-c3-0-63
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  + 5.09·2-s + 17.9·4-s − 18.5·5-s + 50.5·8-s − 94.7·10-s − 1.05·11-s − 58.7·13-s + 113.·16-s + 43.7·17-s + 131.·19-s − 333.·20-s − 5.37·22-s + 161.·23-s + 220.·25-s − 299.·26-s + 64.0·29-s + 55.9·31-s + 175.·32-s + 222.·34-s + 296.·37-s + 671.·38-s − 940.·40-s + 80.6·41-s − 134.·43-s − 18.9·44-s + 821.·46-s + 233.·47-s + ⋯
L(s)  = 1  + 1.80·2-s + 2.24·4-s − 1.66·5-s + 2.23·8-s − 2.99·10-s − 0.0289·11-s − 1.25·13-s + 1.78·16-s + 0.624·17-s + 1.59·19-s − 3.72·20-s − 0.0521·22-s + 1.46·23-s + 1.76·25-s − 2.25·26-s + 0.409·29-s + 0.324·31-s + 0.971·32-s + 1.12·34-s + 1.31·37-s + 2.86·38-s − 3.71·40-s + 0.307·41-s − 0.476·43-s − 0.0648·44-s + 2.63·46-s + 0.724·47-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 5.1516796825.151679682
L(12)L(\frac12) ≈\approx 5.1516796825.151679682
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 1−5.09T+8T2 1 - 5.09T + 8T^{2}
5 1+18.5T+125T2 1 + 18.5T + 125T^{2}
11 1+1.05T+1.33e3T2 1 + 1.05T + 1.33e3T^{2}
13 1+58.7T+2.19e3T2 1 + 58.7T + 2.19e3T^{2}
17 1−43.7T+4.91e3T2 1 - 43.7T + 4.91e3T^{2}
19 1−131.T+6.85e3T2 1 - 131.T + 6.85e3T^{2}
23 1−161.T+1.21e4T2 1 - 161.T + 1.21e4T^{2}
29 1−64.0T+2.43e4T2 1 - 64.0T + 2.43e4T^{2}
31 1−55.9T+2.97e4T2 1 - 55.9T + 2.97e4T^{2}
37 1−296.T+5.06e4T2 1 - 296.T + 5.06e4T^{2}
41 1−80.6T+6.89e4T2 1 - 80.6T + 6.89e4T^{2}
43 1+134.T+7.95e4T2 1 + 134.T + 7.95e4T^{2}
47 1−233.T+1.03e5T2 1 - 233.T + 1.03e5T^{2}
53 1−387.T+1.48e5T2 1 - 387.T + 1.48e5T^{2}
59 1−722.T+2.05e5T2 1 - 722.T + 2.05e5T^{2}
61 1+388.T+2.26e5T2 1 + 388.T + 2.26e5T^{2}
67 1+730.T+3.00e5T2 1 + 730.T + 3.00e5T^{2}
71 1+685.T+3.57e5T2 1 + 685.T + 3.57e5T^{2}
73 1−275.T+3.89e5T2 1 - 275.T + 3.89e5T^{2}
79 1−854.T+4.93e5T2 1 - 854.T + 4.93e5T^{2}
83 1−922.T+5.71e5T2 1 - 922.T + 5.71e5T^{2}
89 1+301.T+7.04e5T2 1 + 301.T + 7.04e5T^{2}
97 1+913.T+9.12e5T2 1 + 913.T + 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

−9.282517451975435014638674012148, −7.950882383811106001454609044179, −7.38014733470074022132095539442, −6.85270039733057877670900319529, −5.55184639852496652550641973113, −4.87613024791808590327393997607, −4.19530409908093360024800364369, −3.26444592284693035249632935752, −2.71002204849745509345082157547, −0.876891727874681402328249459012, 0.876891727874681402328249459012, 2.71002204849745509345082157547, 3.26444592284693035249632935752, 4.19530409908093360024800364369, 4.87613024791808590327393997607, 5.55184639852496652550641973113, 6.85270039733057877670900319529, 7.38014733470074022132095539442, 7.950882383811106001454609044179, 9.282517451975435014638674012148

Graph of the ZZ-function along the critical line