↑

Properties

Label 2-1512-168.83-c0-0-2
Degree 22
Conductor 15121512
Sign 11
Analytic cond. 0.7545860.754586
Root an. cond. 0.8686690.868669
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 7-s + 8-s + 13-s − 14-s + 16-s + 17-s − 23-s + 25-s + 26-s − 28-s − 29-s + 31-s + 32-s + 34-s − 2·41-s − 43-s − 46-s + 49-s + 50-s + 52-s − 53-s − 56-s − 58-s + 59-s − 2·61-s + ⋯
L(s)  = 1  + 2-s + 4-s − 7-s + 8-s + 13-s − 14-s + 16-s + 17-s − 23-s + 25-s + 26-s − 28-s − 29-s + 31-s + 32-s + 34-s − 2·41-s − 43-s − 46-s + 49-s + 50-s + 52-s − 53-s − 56-s − 58-s + 59-s − 2·61-s + ⋯

Functional equation

Λ(s)=(1512s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(1512s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15121512    =    23⋅33⋅72^{3} \cdot 3^{3} \cdot 7
Sign: 11
Analytic conductor: 0.7545860.754586
Root analytic conductor: 0.8686690.868669
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ1512(755,⋅)\chi_{1512} (755, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1512, ( :0), 1)(2,\ 1512,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9921681401.992168140
L(12)L(\frac12) ≈\approx 1.9921681401.992168140
L(1)L(1) 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 1
7 1+T 1 + T
good5 (1−T)(1+T) ( 1 - T )( 1 + T )
11 (1−T)(1+T) ( 1 - T )( 1 + T )
13 1−T+T2 1 - T + T^{2}
17 1−T+T2 1 - T + T^{2}
19 (1−T)(1+T) ( 1 - T )( 1 + T )
23 1+T+T2 1 + T + T^{2}
29 1+T+T2 1 + T + T^{2}
31 1−T+T2 1 - T + T^{2}
37 (1−T)(1+T) ( 1 - T )( 1 + T )
41 (1+T)2 ( 1 + T )^{2}
43 1+T+T2 1 + T + T^{2}
47 (1−T)(1+T) ( 1 - T )( 1 + T )
53 1+T+T2 1 + T + T^{2}
59 1−T+T2 1 - T + T^{2}
61 (1+T)2 ( 1 + T )^{2}
67 1+T+T2 1 + T + T^{2}
71 1+T+T2 1 + T + T^{2}
73 (1−T)(1+T) ( 1 - T )( 1 + T )
79 (1−T)(1+T) ( 1 - T )( 1 + T )
83 (1+T)2 ( 1 + T )^{2}
89 1−T+T2 1 - T + T^{2}
97 (1−T)(1+T) ( 1 - T )( 1 + T )
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

−9.976177758117460441739952635687, −8.810055757627786735145184214051, −7.925871014580012675794904153952, −6.99018334114702224439697050529, −6.25634748728377969491617849342, −5.66066818494798845587028740377, −4.60168303389157687301077617621, −3.54980560312016193768741864073, −3.04239986710352024584250206573, −1.57690789820271454898526195632, 1.57690789820271454898526195632, 3.04239986710352024584250206573, 3.54980560312016193768741864073, 4.60168303389157687301077617621, 5.66066818494798845587028740377, 6.25634748728377969491617849342, 6.99018334114702224439697050529, 7.925871014580012675794904153952, 8.810055757627786735145184214051, 9.976177758117460441739952635687

Graph of the ZZ-function along the critical line