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Properties

Label 2-2160-15.14-c0-0-3
Degree 22
Conductor 21602160
Sign 11
Analytic cond. 1.077981.07798
Root an. cond. 1.038251.03825
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 17-s + 19-s + 23-s + 25-s + 31-s − 2·47-s + 49-s − 53-s − 61-s + 79-s + 83-s − 85-s + 95-s − 2·107-s − 109-s + 2·113-s + 115-s + ⋯
L(s)  = 1  + 5-s − 17-s + 19-s + 23-s + 25-s + 31-s − 2·47-s + 49-s − 53-s − 61-s + 79-s + 83-s − 85-s + 95-s − 2·107-s − 109-s + 2·113-s + 115-s + ⋯

Functional equation

Λ(s)=(2160s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(2160s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21602160    =    24⋅33⋅52^{4} \cdot 3^{3} \cdot 5
Sign: 11
Analytic conductor: 1.077981.07798
Root analytic conductor: 1.038251.03825
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ2160(1889,⋅)\chi_{2160} (1889, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2160, ( :0), 1)(2,\ 2160,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4306078461.430607846
L(12)L(\frac12) ≈\approx 1.4306078461.430607846
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 1
3 1 1
5 1−T 1 - T
good7 (1−T)(1+T) ( 1 - T )( 1 + T )
11 (1−T)(1+T) ( 1 - T )( 1 + T )
13 (1−T)(1+T) ( 1 - T )( 1 + T )
17 1+T+T2 1 + T + T^{2}
19 1−T+T2 1 - T + T^{2}
23 1−T+T2 1 - T + T^{2}
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 1−T+T2 1 - T + T^{2}
37 (1−T)(1+T) ( 1 - T )( 1 + T )
41 (1−T)(1+T) ( 1 - T )( 1 + T )
43 (1−T)(1+T) ( 1 - T )( 1 + T )
47 (1+T)2 ( 1 + T )^{2}
53 1+T+T2 1 + T + T^{2}
59 (1−T)(1+T) ( 1 - T )( 1 + T )
61 1+T+T2 1 + T + T^{2}
67 (1−T)(1+T) ( 1 - T )( 1 + T )
71 (1−T)(1+T) ( 1 - T )( 1 + T )
73 (1−T)(1+T) ( 1 - T )( 1 + T )
79 1−T+T2 1 - T + T^{2}
83 1−T+T2 1 - T + T^{2}
89 (1−T)(1+T) ( 1 - T )( 1 + T )
97 (1−T)(1+T) ( 1 - T )( 1 + T )
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   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

−9.314755982477054221143082758028, −8.646261674320724763652826765045, −7.70301884847421054867270243950, −6.76350400436940707631919213332, −6.22901279508452714497602617344, −5.22337244520440055754597376928, −4.64098926891498131804169404996, −3.31321653895451706956557173084, −2.44418642050525191040624672666, −1.30675294302360740766815383931, 1.30675294302360740766815383931, 2.44418642050525191040624672666, 3.31321653895451706956557173084, 4.64098926891498131804169404996, 5.22337244520440055754597376928, 6.22901279508452714497602617344, 6.76350400436940707631919213332, 7.70301884847421054867270243950, 8.646261674320724763652826765045, 9.314755982477054221143082758028

Graph of the ZZ-function along the critical line