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Properties

Label 2-12e3-1.1-c1-0-13
Degree 22
Conductor 17281728
Sign 11
Analytic cond. 13.798113.7981
Root an. cond. 3.714583.71458
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·7-s + 7·13-s + 19-s − 5·25-s − 4·31-s + 37-s − 8·43-s + 18·49-s + 13·61-s − 11·67-s + 17·73-s − 13·79-s + 35·91-s + 5·97-s − 7·103-s − 2·109-s + ⋯
L(s)  = 1  + 1.88·7-s + 1.94·13-s + 0.229·19-s − 25-s − 0.718·31-s + 0.164·37-s − 1.21·43-s + 18/7·49-s + 1.66·61-s − 1.34·67-s + 1.98·73-s − 1.46·79-s + 3.66·91-s + 0.507·97-s − 0.689·103-s − 0.191·109-s + ⋯

Functional equation

Λ(s)=(1728s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1728s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17281728    =    26⋅332^{6} \cdot 3^{3}
Sign: 11
Analytic conductor: 13.798113.7981
Root analytic conductor: 3.714583.71458
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1728, ( :1/2), 1)(2,\ 1728,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.3608442972.360844297
L(12)L(\frac12) ≈\approx 2.3608442972.360844297
L(32)L(\frac{3}{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)
bad2 1 1
3 1 1
good5 1+pT2 1 + p T^{2}
7 1−5T+pT2 1 - 5 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−7T+pT2 1 - 7 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−T+pT2 1 - T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−T+pT2 1 - T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−13T+pT2 1 - 13 T + p T^{2}
67 1+11T+pT2 1 + 11 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−17T+pT2 1 - 17 T + p T^{2}
79 1+13T+pT2 1 + 13 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−5T+pT2 1 - 5 T + p T^{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

−9.125589838381877994041862400986, −8.360413074571016534853094652479, −8.001529503824122494641395863146, −7.02193571198558069600312241264, −5.94215330426161837619813679812, −5.28577670060288457890184502723, −4.31915400924746551561976811972, −3.53995560336243251785891738493, −2.01230440740899716446600775646, −1.20941721581339133701993176268, 1.20941721581339133701993176268, 2.01230440740899716446600775646, 3.53995560336243251785891738493, 4.31915400924746551561976811972, 5.28577670060288457890184502723, 5.94215330426161837619813679812, 7.02193571198558069600312241264, 8.001529503824122494641395863146, 8.360413074571016534853094652479, 9.125589838381877994041862400986

Graph of the ZZ-function along the critical line