↑

Properties

Label 2-4650-1.1-c1-0-20
Degree 22
Conductor 46504650
Sign 11
Analytic cond. 37.130437.1304
Root an. cond. 6.093476.09347
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  − 2-s − 3-s + 4-s + 6-s + 5·7-s − 8-s + 9-s − 11-s − 12-s − 5·14-s + 16-s − 4·17-s − 18-s + 3·19-s − 5·21-s + 22-s − 23-s + 24-s − 27-s + 5·28-s + 6·29-s − 31-s − 32-s + 33-s + 4·34-s + 36-s − 4·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.88·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 1.33·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.688·19-s − 1.09·21-s + 0.213·22-s − 0.208·23-s + 0.204·24-s − 0.192·27-s + 0.944·28-s + 1.11·29-s − 0.179·31-s − 0.176·32-s + 0.174·33-s + 0.685·34-s + 1/6·36-s − 0.657·37-s + ⋯

Functional equation

Λ(s)=(4650s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(4650s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 46504650    =    2⋅3⋅52⋅312 \cdot 3 \cdot 5^{2} \cdot 31
Sign: 11
Analytic conductor: 37.130437.1304
Root analytic conductor: 6.093476.09347
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 4650, ( :1/2), 1)(2,\ 4650,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4017812461.401781246
L(12)L(\frac12) ≈\approx 1.4017812461.401781246
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+T 1 + T
5 1 1
31 1+T 1 + T
good7 1−5T+pT2 1 - 5 T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1−3T+pT2 1 - 3 T + p T^{2}
23 1+T+pT2 1 + T + p T^{2}
29 1−6T+pT2 1 - 6 T + p T^{2}
37 1+4T+pT2 1 + 4 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1−3T+pT2 1 - 3 T + p T^{2}
59 1+14T+pT2 1 + 14 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1−9T+pT2 1 - 9 T + p T^{2}
73 1+7T+pT2 1 + 7 T + p T^{2}
79 1−15T+pT2 1 - 15 T + p T^{2}
83 1+10T+pT2 1 + 10 T + p T^{2}
89 1+T+pT2 1 + T + p T^{2}
97 1−10T+pT2 1 - 10 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

−8.324144007286194476239608201124, −7.65953445349591187460531797012, −7.05127744977499064075254649470, −6.19342098565817811456718165780, −5.26005152525608288771498772826, −4.82598050501665912095185675835, −3.91147502021945010746086829763, −2.51938515684160805822243348837, −1.72974318761389879716166327223, −0.78630454339232826724681158124, 0.78630454339232826724681158124, 1.72974318761389879716166327223, 2.51938515684160805822243348837, 3.91147502021945010746086829763, 4.82598050501665912095185675835, 5.26005152525608288771498772826, 6.19342098565817811456718165780, 7.05127744977499064075254649470, 7.65953445349591187460531797012, 8.324144007286194476239608201124

Graph of the ZZ-function along the critical line