↑

Properties

Label 2-4788-1.1-c1-0-9
Degree 22
Conductor 47884788
Sign 11
Analytic cond. 38.232338.2323
Root an. cond. 6.183236.18323
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.29·5-s + 7-s − 2·11-s + 5.71·13-s − 1.29·17-s − 19-s − 2·23-s − 3.31·25-s + 10.7·29-s − 3.71·31-s − 1.29·35-s − 0.599·37-s + 4·41-s + 10.3·43-s − 10.1·47-s + 49-s − 0.700·53-s + 2.59·55-s + 3.40·61-s − 7.42·65-s − 2.59·67-s + 0.700·71-s + 13.4·73-s − 2·77-s − 6.02·79-s + 12.7·83-s + 1.68·85-s + ⋯
L(s)  = 1  − 0.581·5-s + 0.377·7-s − 0.603·11-s + 1.58·13-s − 0.315·17-s − 0.229·19-s − 0.417·23-s − 0.662·25-s + 1.99·29-s − 0.666·31-s − 0.219·35-s − 0.0985·37-s + 0.624·41-s + 1.57·43-s − 1.47·47-s + 0.142·49-s − 0.0961·53-s + 0.350·55-s + 0.435·61-s − 0.920·65-s − 0.317·67-s + 0.0831·71-s + 1.57·73-s − 0.227·77-s − 0.677·79-s + 1.39·83-s + 0.183·85-s + ⋯

Functional equation

Λ(s)=(4788s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4788s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 47884788    =    22⋅32⋅7⋅192^{2} \cdot 3^{2} \cdot 7 \cdot 19
Sign: 11
Analytic conductor: 38.232338.2323
Root analytic conductor: 6.183236.18323
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4788, ( :1/2), 1)(2,\ 4788,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7272201691.727220169
L(12)L(\frac12) ≈\approx 1.7272201691.727220169
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+1.29T+5T2 1 + 1.29T + 5T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
13 1−5.71T+13T2 1 - 5.71T + 13T^{2}
17 1+1.29T+17T2 1 + 1.29T + 17T^{2}
23 1+2T+23T2 1 + 2T + 23T^{2}
29 1−10.7T+29T2 1 - 10.7T + 29T^{2}
31 1+3.71T+31T2 1 + 3.71T + 31T^{2}
37 1+0.599T+37T2 1 + 0.599T + 37T^{2}
41 1−4T+41T2 1 - 4T + 41T^{2}
43 1−10.3T+43T2 1 - 10.3T + 43T^{2}
47 1+10.1T+47T2 1 + 10.1T + 47T^{2}
53 1+0.700T+53T2 1 + 0.700T + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−3.40T+61T2 1 - 3.40T + 61T^{2}
67 1+2.59T+67T2 1 + 2.59T + 67T^{2}
71 1−0.700T+71T2 1 - 0.700T + 71T^{2}
73 1−13.4T+73T2 1 - 13.4T + 73T^{2}
79 1+6.02T+79T2 1 + 6.02T + 79T^{2}
83 1−12.7T+83T2 1 - 12.7T + 83T^{2}
89 1−4T+89T2 1 - 4T + 89T^{2}
97 1+2T+97T2 1 + 2T + 97T^{2}
show more
show less
   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

−8.139254648520164096402630087330, −7.84207010344993905814159649291, −6.78198260623656034077653943532, −6.15378104366534507591352389367, −5.36870529312430896292562431256, −4.43992733834217977085799158121, −3.85133339867108035484888162138, −2.94396996101143365399959939923, −1.88759697978961040943738250722, −0.73338856144486033953605719948, 0.73338856144486033953605719948, 1.88759697978961040943738250722, 2.94396996101143365399959939923, 3.85133339867108035484888162138, 4.43992733834217977085799158121, 5.36870529312430896292562431256, 6.15378104366534507591352389367, 6.78198260623656034077653943532, 7.84207010344993905814159649291, 8.139254648520164096402630087330

Graph of the ZZ-function along the critical line