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Properties

Label 2-9576-1.1-c1-0-39
Degree 22
Conductor 95769576
Sign 11
Analytic cond. 76.464776.4647
Root an. cond. 8.744418.74441
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  + 1.23·5-s − 7-s + 5.23·11-s − 0.763·13-s − 7.23·17-s + 19-s + 3.23·23-s − 3.47·25-s − 8.47·29-s + 0.472·31-s − 1.23·35-s + 8.94·37-s − 2·41-s − 4·43-s + 6.47·47-s + 49-s + 0.472·53-s + 6.47·55-s + 8·59-s + 8.47·61-s − 0.944·65-s + 0.763·67-s + 1.52·71-s + 4.47·73-s − 5.23·77-s + 15.7·79-s − 2·83-s + ⋯
L(s)  = 1  + 0.552·5-s − 0.377·7-s + 1.57·11-s − 0.211·13-s − 1.75·17-s + 0.229·19-s + 0.674·23-s − 0.694·25-s − 1.57·29-s + 0.0847·31-s − 0.208·35-s + 1.47·37-s − 0.312·41-s − 0.609·43-s + 0.944·47-s + 0.142·49-s + 0.0648·53-s + 0.872·55-s + 1.04·59-s + 1.08·61-s − 0.117·65-s + 0.0933·67-s + 0.181·71-s + 0.523·73-s − 0.596·77-s + 1.76·79-s − 0.219·83-s + ⋯

Functional equation

Λ(s)=(9576s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9576s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 95769576    =    23⋅32⋅7⋅192^{3} \cdot 3^{2} \cdot 7 \cdot 19
Sign: 11
Analytic conductor: 76.464776.4647
Root analytic conductor: 8.744418.74441
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9576, ( :1/2), 1)(2,\ 9576,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1942950922.194295092
L(12)L(\frac12) ≈\approx 2.1942950922.194295092
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
7 1+T 1 + T
19 1−T 1 - T
good5 1−1.23T+5T2 1 - 1.23T + 5T^{2}
11 1−5.23T+11T2 1 - 5.23T + 11T^{2}
13 1+0.763T+13T2 1 + 0.763T + 13T^{2}
17 1+7.23T+17T2 1 + 7.23T + 17T^{2}
23 1−3.23T+23T2 1 - 3.23T + 23T^{2}
29 1+8.47T+29T2 1 + 8.47T + 29T^{2}
31 1−0.472T+31T2 1 - 0.472T + 31T^{2}
37 1−8.94T+37T2 1 - 8.94T + 37T^{2}
41 1+2T+41T2 1 + 2T + 41T^{2}
43 1+4T+43T2 1 + 4T + 43T^{2}
47 1−6.47T+47T2 1 - 6.47T + 47T^{2}
53 1−0.472T+53T2 1 - 0.472T + 53T^{2}
59 1−8T+59T2 1 - 8T + 59T^{2}
61 1−8.47T+61T2 1 - 8.47T + 61T^{2}
67 1−0.763T+67T2 1 - 0.763T + 67T^{2}
71 1−1.52T+71T2 1 - 1.52T + 71T^{2}
73 1−4.47T+73T2 1 - 4.47T + 73T^{2}
79 1−15.7T+79T2 1 - 15.7T + 79T^{2}
83 1+2T+83T2 1 + 2T + 83T^{2}
89 1−10.9T+89T2 1 - 10.9T + 89T^{2}
97 1−1.23T+97T2 1 - 1.23T + 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

−7.55695829701518011535743341331, −6.84158340841337677529916553504, −6.42050509685313391210844701869, −5.76824538076009081581471509434, −4.92582765674797315601368394393, −4.08577905614069686340656597051, −3.59705079604356304886173667533, −2.42948670189715872435204141802, −1.84007508218933729871461112504, −0.70503064142347803198283574142, 0.70503064142347803198283574142, 1.84007508218933729871461112504, 2.42948670189715872435204141802, 3.59705079604356304886173667533, 4.08577905614069686340656597051, 4.92582765674797315601368394393, 5.76824538076009081581471509434, 6.42050509685313391210844701869, 6.84158340841337677529916553504, 7.55695829701518011535743341331

Graph of the ZZ-function along the critical line