↑

Properties

Label 2-4788-1.1-c1-0-18
Degree 22
Conductor 47884788
Sign 11
Analytic cond. 38.232338.2323
Root an. cond. 6.183236.18323
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 7-s + 2·11-s + 6·13-s + 8·17-s + 19-s + 2·23-s − 5·25-s − 6·29-s + 10·37-s + 8·41-s − 12·43-s − 4·47-s + 49-s − 2·53-s + 8·59-s − 10·61-s − 12·67-s + 10·71-s − 2·73-s − 2·77-s − 8·79-s + 4·83-s + 8·89-s − 6·91-s + 2·97-s + 4·101-s − 2·107-s + ⋯
L(s)  = 1  − 0.377·7-s + 0.603·11-s + 1.66·13-s + 1.94·17-s + 0.229·19-s + 0.417·23-s − 25-s − 1.11·29-s + 1.64·37-s + 1.24·41-s − 1.82·43-s − 0.583·47-s + 1/7·49-s − 0.274·53-s + 1.04·59-s − 1.28·61-s − 1.46·67-s + 1.18·71-s − 0.234·73-s − 0.227·77-s − 0.900·79-s + 0.439·83-s + 0.847·89-s − 0.628·91-s + 0.203·97-s + 0.398·101-s − 0.193·107-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: yes
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 2.2766810532.276681053
L(12)L(\frac12) ≈\approx 2.2766810532.276681053
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+pT2 1 + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1−8T+pT2 1 - 8 T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−8T+pT2 1 - 8 T + p T^{2}
43 1+12T+pT2 1 + 12 T + p T^{2}
47 1+4T+pT2 1 + 4 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+8T+pT2 1 + 8 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−8T+pT2 1 - 8 T + p T^{2}
97 1−2T+pT2 1 - 2 T + p T^{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

−8.141126835189670259468431790625, −7.71179363869623773766951421262, −6.78425966044936463996649660556, −5.92461069948544809607605082020, −5.68657546334957466916060281266, −4.45128064822373452531412119614, −3.58432542369787507380741066676, −3.17785935881446270719128297999, −1.73108969966905643937793181595, −0.895071874805042573229366117238, 0.895071874805042573229366117238, 1.73108969966905643937793181595, 3.17785935881446270719128297999, 3.58432542369787507380741066676, 4.45128064822373452531412119614, 5.68657546334957466916060281266, 5.92461069948544809607605082020, 6.78425966044936463996649660556, 7.71179363869623773766951421262, 8.141126835189670259468431790625

Graph of the ZZ-function along the critical line