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Properties

Label 2-1134-1.1-c1-0-20
Degree 22
Conductor 11341134
Sign −1-1
Analytic cond. 9.055039.05503
Root an. cond. 3.009153.00915
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 3.73·5-s − 7-s + 8-s − 3.73·10-s + 4.19·11-s + 0.464·13-s − 14-s + 16-s − 7·17-s − 2.73·19-s − 3.73·20-s + 4.19·22-s − 6.19·23-s + 8.92·25-s + 0.464·26-s − 28-s − 8.46·29-s − 2.19·31-s + 32-s − 7·34-s + 3.73·35-s − 6.66·37-s − 2.73·38-s − 3.73·40-s − 9.46·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.66·5-s − 0.377·7-s + 0.353·8-s − 1.18·10-s + 1.26·11-s + 0.128·13-s − 0.267·14-s + 0.250·16-s − 1.69·17-s − 0.626·19-s − 0.834·20-s + 0.894·22-s − 1.29·23-s + 1.78·25-s + 0.0910·26-s − 0.188·28-s − 1.57·29-s − 0.394·31-s + 0.176·32-s − 1.20·34-s + 0.630·35-s − 1.09·37-s − 0.443·38-s − 0.590·40-s − 1.47·41-s + ⋯

Functional equation

Λ(s)=(1134s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1134s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11341134    =    2⋅34⋅72 \cdot 3^{4} \cdot 7
Sign: −1-1
Analytic conductor: 9.055039.05503
Root analytic conductor: 3.009153.00915
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1134, ( :1/2), −1)(2,\ 1134,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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−T 1 - T
3 1 1
7 1+T 1 + T
good5 1+3.73T+5T2 1 + 3.73T + 5T^{2}
11 1−4.19T+11T2 1 - 4.19T + 11T^{2}
13 1−0.464T+13T2 1 - 0.464T + 13T^{2}
17 1+7T+17T2 1 + 7T + 17T^{2}
19 1+2.73T+19T2 1 + 2.73T + 19T^{2}
23 1+6.19T+23T2 1 + 6.19T + 23T^{2}
29 1+8.46T+29T2 1 + 8.46T + 29T^{2}
31 1+2.19T+31T2 1 + 2.19T + 31T^{2}
37 1+6.66T+37T2 1 + 6.66T + 37T^{2}
41 1+9.46T+41T2 1 + 9.46T + 41T^{2}
43 1−5.46T+43T2 1 - 5.46T + 43T^{2}
47 1+1.26T+47T2 1 + 1.26T + 47T^{2}
53 1−2.53T+53T2 1 - 2.53T + 53T^{2}
59 1−6.19T+59T2 1 - 6.19T + 59T^{2}
61 1+9.92T+61T2 1 + 9.92T + 61T^{2}
67 1+3.26T+67T2 1 + 3.26T + 67T^{2}
71 1+13.4T+71T2 1 + 13.4T + 71T^{2}
73 1−11.7T+73T2 1 - 11.7T + 73T^{2}
79 1−15.1T+79T2 1 - 15.1T + 79T^{2}
83 1−14.5T+83T2 1 - 14.5T + 83T^{2}
89 1−3.92T+89T2 1 - 3.92T + 89T^{2}
97 1+2.92T+97T2 1 + 2.92T + 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

−9.207135692349532420199214553518, −8.567860918077030616389573513678, −7.58938048544217216140004445826, −6.83580903220873777734978824021, −6.17325726335194061442348403962, −4.79036484471245917661873888650, −3.91150210175583802167136505912, −3.61666009803687336262678884369, −2.00739084460189131272843904705, 0, 2.00739084460189131272843904705, 3.61666009803687336262678884369, 3.91150210175583802167136505912, 4.79036484471245917661873888650, 6.17325726335194061442348403962, 6.83580903220873777734978824021, 7.58938048544217216140004445826, 8.567860918077030616389573513678, 9.207135692349532420199214553518

Graph of the ZZ-function along the critical line