↑

Properties

Label 2-3240-40.19-c0-0-1
Degree 22
Conductor 32403240
Sign 11
Analytic cond. 1.616971.61697
Root an. cond. 1.271601.27160
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 1.73·7-s − 8-s − 10-s + 1.73·13-s + 1.73·14-s + 16-s + 19-s + 20-s − 23-s + 25-s − 1.73·26-s − 1.73·28-s − 32-s − 1.73·35-s − 38-s − 40-s − 1.73·41-s + 46-s + 47-s + 1.99·49-s − 50-s + 1.73·52-s + 53-s + 1.73·56-s + ⋯
L(s)  = 1  − 2-s + 4-s + 5-s − 1.73·7-s − 8-s − 10-s + 1.73·13-s + 1.73·14-s + 16-s + 19-s + 20-s − 23-s + 25-s − 1.73·26-s − 1.73·28-s − 32-s − 1.73·35-s − 38-s − 40-s − 1.73·41-s + 46-s + 47-s + 1.99·49-s − 50-s + 1.73·52-s + 53-s + 1.73·56-s + ⋯

Functional equation

Λ(s)=(3240s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(3240s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32403240    =    23⋅34⋅52^{3} \cdot 3^{4} \cdot 5
Sign: 11
Analytic conductor: 1.616971.61697
Root analytic conductor: 1.271601.27160
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3240(1459,⋅)\chi_{3240} (1459, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3240, ( :0), 1)(2,\ 3240,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.86828247700.8682824770
L(12)L(\frac12) ≈\approx 0.86828247700.8682824770
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
5 1−T 1 - T
good7 1+1.73T+T2 1 + 1.73T + T^{2}
11 1+T2 1 + T^{2}
13 1−1.73T+T2 1 - 1.73T + T^{2}
17 1−T2 1 - T^{2}
19 1−T+T2 1 - T + T^{2}
23 1+T+T2 1 + T + T^{2}
29 1−T2 1 - T^{2}
31 1−T2 1 - T^{2}
37 1+T2 1 + T^{2}
41 1+1.73T+T2 1 + 1.73T + T^{2}
43 1−T2 1 - T^{2}
47 1−T+T2 1 - T + T^{2}
53 1−T+T2 1 - T + T^{2}
59 1−1.73T+T2 1 - 1.73T + T^{2}
61 1−T2 1 - T^{2}
67 1−T2 1 - T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1+T2 1 + T^{2}
97 1−T2 1 - 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

−8.897890477540212442963696653360, −8.397375180912137666304748937216, −7.21853567200084208870066654996, −6.61138011113362149162249830833, −6.01154955992664242009573294743, −5.52496360970558548656805260817, −3.74303788260714063539041396806, −3.15290028292440029218077555315, −2.13504834424517437338992909118, −0.978875688099410759521335671866, 0.978875688099410759521335671866, 2.13504834424517437338992909118, 3.15290028292440029218077555315, 3.74303788260714063539041396806, 5.52496360970558548656805260817, 6.01154955992664242009573294743, 6.61138011113362149162249830833, 7.21853567200084208870066654996, 8.397375180912137666304748937216, 8.897890477540212442963696653360

Graph of the ZZ-function along the critical line