↑

Properties

Label 2-9360-1.1-c1-0-35
Degree 22
Conductor 93609360
Sign 11
Analytic cond. 74.739974.7399
Root an. cond. 8.645228.64522
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  + 5-s − 13-s + 6·17-s − 4·23-s + 25-s + 10·29-s − 6·37-s − 2·41-s + 4·43-s − 7·49-s + 6·53-s + 6·61-s − 65-s − 4·67-s + 16·71-s − 2·73-s + 4·83-s + 6·85-s + 6·89-s + 14·97-s − 6·101-s − 12·103-s − 4·107-s − 14·109-s − 10·113-s − 4·115-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.277·13-s + 1.45·17-s − 0.834·23-s + 1/5·25-s + 1.85·29-s − 0.986·37-s − 0.312·41-s + 0.609·43-s − 49-s + 0.824·53-s + 0.768·61-s − 0.124·65-s − 0.488·67-s + 1.89·71-s − 0.234·73-s + 0.439·83-s + 0.650·85-s + 0.635·89-s + 1.42·97-s − 0.597·101-s − 1.18·103-s − 0.386·107-s − 1.34·109-s − 0.940·113-s − 0.373·115-s + ⋯

Functional equation

Λ(s)=(9360s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9360s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 93609360    =    24⋅32⋅5⋅132^{4} \cdot 3^{2} \cdot 5 \cdot 13
Sign: 11
Analytic conductor: 74.739974.7399
Root analytic conductor: 8.645228.64522
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9360, ( :1/2), 1)(2,\ 9360,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.3512590452.351259045
L(12)L(\frac12) ≈\approx 2.3512590452.351259045
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
5 1−T 1 - T
13 1+T 1 + T
good7 1+pT2 1 + p T^{2}
11 1+pT2 1 + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1−10T+pT2 1 - 10 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+6T+pT2 1 + 6 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−6T+pT2 1 - 6 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−16T+pT2 1 - 16 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−14T+pT2 1 - 14 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

−7.86024313710614696532097430664, −6.90823153056793002056497471386, −6.41051413056067833990175014344, −5.55860505151888914144681124469, −5.10592887722971602042874207321, −4.20725491653826331663773790383, −3.38062750090213723463354695459, −2.63741599748730669205673807869, −1.71741878316152659310548007340, −0.75375237820186639289019479057, 0.75375237820186639289019479057, 1.71741878316152659310548007340, 2.63741599748730669205673807869, 3.38062750090213723463354695459, 4.20725491653826331663773790383, 5.10592887722971602042874207321, 5.55860505151888914144681124469, 6.41051413056067833990175014344, 6.90823153056793002056497471386, 7.86024313710614696532097430664

Graph of the ZZ-function along the critical line