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Properties

Label 2-72e2-1.1-c1-0-5
Degree 22
Conductor 51845184
Sign 11
Analytic cond. 41.394441.3944
Root an. cond. 6.433856.43385
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.46·5-s − 0.464·13-s − 7.92·17-s + 14.9·25-s − 8.46·29-s − 11.3·37-s + 10·41-s − 7·49-s − 14·53-s + 5.39·61-s + 2.07·65-s − 10.8·73-s + 35.3·85-s + 8.85·89-s + 18·97-s + 2·101-s + 20.3·109-s − 6.85·113-s + ⋯
L(s)  = 1  − 1.99·5-s − 0.128·13-s − 1.92·17-s + 2.98·25-s − 1.57·29-s − 1.87·37-s + 1.56·41-s − 49-s − 1.92·53-s + 0.690·61-s + 0.256·65-s − 1.27·73-s + 3.83·85-s + 0.938·89-s + 1.82·97-s + 0.199·101-s + 1.94·109-s − 0.644·113-s + ⋯

Functional equation

Λ(s)=(5184s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5184s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 51845184    =    26⋅342^{6} \cdot 3^{4}
Sign: 11
Analytic conductor: 41.394441.3944
Root analytic conductor: 6.433856.43385
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5184, ( :1/2), 1)(2,\ 5184,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.51373315910.5137331591
L(12)L(\frac12) ≈\approx 0.51373315910.5137331591
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+4.46T+5T2 1 + 4.46T + 5T^{2}
7 1+7T2 1 + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+0.464T+13T2 1 + 0.464T + 13T^{2}
17 1+7.92T+17T2 1 + 7.92T + 17T^{2}
19 1+19T2 1 + 19T^{2}
23 1+23T2 1 + 23T^{2}
29 1+8.46T+29T2 1 + 8.46T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+11.3T+37T2 1 + 11.3T + 37T^{2}
41 1−10T+41T2 1 - 10T + 41T^{2}
43 1+43T2 1 + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1+14T+53T2 1 + 14T + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−5.39T+61T2 1 - 5.39T + 61T^{2}
67 1+67T2 1 + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+10.8T+73T2 1 + 10.8T + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1−8.85T+89T2 1 - 8.85T + 89T^{2}
97 1−18T+97T2 1 - 18T + 97T^{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

−8.156612511271434346526020858050, −7.47097474835230567969543299763, −7.00686009611450678944376664071, −6.22022506250851922997472385938, −5.01331836638713343789264024931, −4.44494178778729488061745237176, −3.77906035919093832763527125450, −3.07409752224957339324227664587, −1.90448245573759145325687976885, −0.37288790028450785323453949242, 0.37288790028450785323453949242, 1.90448245573759145325687976885, 3.07409752224957339324227664587, 3.77906035919093832763527125450, 4.44494178778729488061745237176, 5.01331836638713343789264024931, 6.22022506250851922997472385938, 7.00686009611450678944376664071, 7.47097474835230567969543299763, 8.156612511271434346526020858050

Graph of the ZZ-function along the critical line