↑

Properties

Label 2-1323-1.1-c3-0-41
Degree 22
Conductor 13231323
Sign 11
Analytic cond. 78.059578.0595
Root an. cond. 8.835138.83513
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8·4-s + 89·13-s + 64·16-s + 56·19-s − 125·25-s − 289·31-s − 433·37-s + 71·43-s − 712·52-s + 719·61-s − 512·64-s + 1.00e3·67-s + 1.19e3·73-s − 448·76-s + 503·79-s − 523·97-s + 1.00e3·100-s − 19·103-s + 2.21e3·109-s + ⋯
L(s)  = 1  − 4-s + 1.89·13-s + 16-s + 0.676·19-s − 25-s − 1.67·31-s − 1.92·37-s + 0.251·43-s − 1.89·52-s + 1.50·61-s − 64-s + 1.83·67-s + 1.90·73-s − 0.676·76-s + 0.716·79-s − 0.547·97-s + 100-s − 0.0181·103-s + 1.94·109-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 11
Analytic conductor: 78.059578.0595
Root analytic conductor: 8.835138.83513
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1323, ( :3/2), 1)(2,\ 1323,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.6021052601.602105260
L(12)L(\frac12) ≈\approx 1.6021052601.602105260
L(52)L(\frac{5}{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)
bad3 1 1
7 1 1
good2 1+p3T2 1 + p^{3} T^{2}
5 1+p3T2 1 + p^{3} T^{2}
11 1+p3T2 1 + p^{3} T^{2}
13 1−89T+p3T2 1 - 89 T + p^{3} T^{2}
17 1+p3T2 1 + p^{3} T^{2}
19 1−56T+p3T2 1 - 56 T + p^{3} T^{2}
23 1+p3T2 1 + p^{3} T^{2}
29 1+p3T2 1 + p^{3} T^{2}
31 1+289T+p3T2 1 + 289 T + p^{3} T^{2}
37 1+433T+p3T2 1 + 433 T + p^{3} T^{2}
41 1+p3T2 1 + p^{3} T^{2}
43 1−71T+p3T2 1 - 71 T + p^{3} T^{2}
47 1+p3T2 1 + p^{3} T^{2}
53 1+p3T2 1 + p^{3} T^{2}
59 1+p3T2 1 + p^{3} T^{2}
61 1−719T+p3T2 1 - 719 T + p^{3} T^{2}
67 1−1007T+p3T2 1 - 1007 T + p^{3} T^{2}
71 1+p3T2 1 + p^{3} T^{2}
73 1−1190T+p3T2 1 - 1190 T + p^{3} T^{2}
79 1−503T+p3T2 1 - 503 T + p^{3} T^{2}
83 1+p3T2 1 + p^{3} T^{2}
89 1+p3T2 1 + p^{3} T^{2}
97 1+523T+p3T2 1 + 523 T + p^{3} 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

−9.163002706359222015692792715617, −8.548609339522697578618203263630, −7.84571362442804508379736964792, −6.77090679114611187415720676639, −5.73241800004350870429436957795, −5.17350018539273291030212347386, −3.82674854515148048284692065925, −3.54914775531436142843934480700, −1.78034677135038308101569687456, −0.66359926916394254861701684044, 0.66359926916394254861701684044, 1.78034677135038308101569687456, 3.54914775531436142843934480700, 3.82674854515148048284692065925, 5.17350018539273291030212347386, 5.73241800004350870429436957795, 6.77090679114611187415720676639, 7.84571362442804508379736964792, 8.548609339522697578618203263630, 9.163002706359222015692792715617

Graph of the ZZ-function along the critical line