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Properties

Label 2-1872-52.51-c0-0-1
Degree 22
Conductor 18721872
Sign 11
Analytic cond. 0.9342490.934249
Root an. cond. 0.9665650.966565
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  − 13-s + 2·17-s + 25-s + 2·29-s − 49-s − 2·53-s + 2·61-s − 2·101-s + 2·113-s + ⋯
L(s)  = 1  − 13-s + 2·17-s + 25-s + 2·29-s − 49-s − 2·53-s + 2·61-s − 2·101-s + 2·113-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: 11
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ1872(415,⋅)\chi_{1872} (415, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1872, ( :0), 1)(2,\ 1872,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1820174241.182017424
L(12)L(\frac12) ≈\approx 1.1820174241.182017424
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
13 1+T 1 + T
good5 (1−T)(1+T) ( 1 - T )( 1 + T )
7 1+T2 1 + T^{2}
11 1+T2 1 + T^{2}
17 (1−T)2 ( 1 - T )^{2}
19 1+T2 1 + T^{2}
23 (1−T)(1+T) ( 1 - T )( 1 + T )
29 (1−T)2 ( 1 - T )^{2}
31 1+T2 1 + 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+T2 1 + T^{2}
53 (1+T)2 ( 1 + T )^{2}
59 1+T2 1 + T^{2}
61 (1−T)2 ( 1 - T )^{2}
67 1+T2 1 + T^{2}
71 1+T2 1 + T^{2}
73 (1−T)(1+T) ( 1 - T )( 1 + T )
79 (1−T)(1+T) ( 1 - T )( 1 + T )
83 1+T2 1 + 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

−9.656955690253039770308071705721, −8.509244514392409736767076060643, −7.907418238542288715463769777006, −7.08906119093741135118384416472, −6.28888456993578558991317303464, −5.26289914734159044212161307119, −4.68037945037740256630262234013, −3.41668235937234969096213356514, −2.64732708754186443775809146998, −1.18285865140344046637603977780, 1.18285865140344046637603977780, 2.64732708754186443775809146998, 3.41668235937234969096213356514, 4.68037945037740256630262234013, 5.26289914734159044212161307119, 6.28888456993578558991317303464, 7.08906119093741135118384416472, 7.907418238542288715463769777006, 8.509244514392409736767076060643, 9.656955690253039770308071705721

Graph of the ZZ-function along the critical line