↑

Properties

Label 2-4788-1.1-c1-0-5
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  + 7-s − 3.46·11-s − 4·13-s − 6.92·17-s + 19-s − 3.46·23-s − 5·25-s + 3.46·29-s + 8·31-s + 2·37-s + 6.92·41-s + 8·43-s − 3.46·47-s + 49-s + 10.3·53-s + 13.8·59-s + 2·61-s + 2·67-s − 3.46·71-s + 14·73-s − 3.46·77-s + 14·79-s + 10.3·83-s − 13.8·89-s − 4·91-s + 8·97-s − 4·103-s + ⋯
L(s)  = 1  + 0.377·7-s − 1.04·11-s − 1.10·13-s − 1.68·17-s + 0.229·19-s − 0.722·23-s − 25-s + 0.643·29-s + 1.43·31-s + 0.328·37-s + 1.08·41-s + 1.21·43-s − 0.505·47-s + 0.142·49-s + 1.42·53-s + 1.80·59-s + 0.256·61-s + 0.244·67-s − 0.411·71-s + 1.63·73-s − 0.394·77-s + 1.57·79-s + 1.14·83-s − 1.46·89-s − 0.419·91-s + 0.812·97-s − 0.394·103-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 1.3950341551.395034155
L(12)L(\frac12) ≈\approx 1.3950341551.395034155
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+5T2 1 + 5T^{2}
11 1+3.46T+11T2 1 + 3.46T + 11T^{2}
13 1+4T+13T2 1 + 4T + 13T^{2}
17 1+6.92T+17T2 1 + 6.92T + 17T^{2}
23 1+3.46T+23T2 1 + 3.46T + 23T^{2}
29 1−3.46T+29T2 1 - 3.46T + 29T^{2}
31 1−8T+31T2 1 - 8T + 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1−6.92T+41T2 1 - 6.92T + 41T^{2}
43 1−8T+43T2 1 - 8T + 43T^{2}
47 1+3.46T+47T2 1 + 3.46T + 47T^{2}
53 1−10.3T+53T2 1 - 10.3T + 53T^{2}
59 1−13.8T+59T2 1 - 13.8T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1−2T+67T2 1 - 2T + 67T^{2}
71 1+3.46T+71T2 1 + 3.46T + 71T^{2}
73 1−14T+73T2 1 - 14T + 73T^{2}
79 1−14T+79T2 1 - 14T + 79T^{2}
83 1−10.3T+83T2 1 - 10.3T + 83T^{2}
89 1+13.8T+89T2 1 + 13.8T + 89T^{2}
97 1−8T+97T2 1 - 8T + 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

−8.146126621577866774811944768055, −7.68605915912554346282636713151, −6.88363124973409026553181559355, −6.11596017977960070102375215482, −5.25436758350431740350857426748, −4.61315100650871195355261603975, −3.92025909385226681172064085787, −2.44080547567522048938679262169, −2.33846848866104011986554243357, −0.61990338528487705978051009426, 0.61990338528487705978051009426, 2.33846848866104011986554243357, 2.44080547567522048938679262169, 3.92025909385226681172064085787, 4.61315100650871195355261603975, 5.25436758350431740350857426748, 6.11596017977960070102375215482, 6.88363124973409026553181559355, 7.68605915912554346282636713151, 8.146126621577866774811944768055

Graph of the ZZ-function along the critical line