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Properties

Label 2-1920-120.29-c0-0-6
Degree 22
Conductor 19201920
Sign 11
Analytic cond. 0.9582040.958204
Root an. cond. 0.9788790.978879
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  + 3-s + 5-s + 9-s + 15-s − 2·23-s + 25-s + 27-s − 2·29-s − 2·43-s + 45-s + 2·47-s + 49-s − 2·67-s − 2·69-s + 75-s + 81-s − 2·87-s + 2·101-s − 2·115-s + ⋯
L(s)  = 1  + 3-s + 5-s + 9-s + 15-s − 2·23-s + 25-s + 27-s − 2·29-s − 2·43-s + 45-s + 2·47-s + 49-s − 2·67-s − 2·69-s + 75-s + 81-s − 2·87-s + 2·101-s − 2·115-s + ⋯

Functional equation

Λ(s)=(1920s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(1920s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19201920    =    27⋅3⋅52^{7} \cdot 3 \cdot 5
Sign: 11
Analytic conductor: 0.9582040.958204
Root analytic conductor: 0.9788790.978879
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ1920(449,⋅)\chi_{1920} (449, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1920, ( :0), 1)(2,\ 1920,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8850093231.885009323
L(12)L(\frac12) ≈\approx 1.8850093231.885009323
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−T 1 - T
5 1−T 1 - T
good7 (1−T)(1+T) ( 1 - T )( 1 + T )
11 1+T2 1 + T^{2}
13 1+T2 1 + T^{2}
17 1+T2 1 + T^{2}
19 (1−T)(1+T) ( 1 - T )( 1 + T )
23 (1+T)2 ( 1 + T )^{2}
29 (1+T)2 ( 1 + T )^{2}
31 1+T2 1 + T^{2}
37 1+T2 1 + T^{2}
41 (1−T)(1+T) ( 1 - T )( 1 + T )
43 (1+T)2 ( 1 + T )^{2}
47 (1−T)2 ( 1 - T )^{2}
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 1+T2 1 + T^{2}
61 (1−T)(1+T) ( 1 - T )( 1 + T )
67 (1+T)2 ( 1 + T )^{2}
71 (1−T)(1+T) ( 1 - T )( 1 + T )
73 (1−T)(1+T) ( 1 - T )( 1 + T )
79 1+T2 1 + T^{2}
83 (1−T)(1+T) ( 1 - T )( 1 + T )
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

−9.292538845169000671451836209939, −8.780910300492600097267113344300, −7.85564758843381750571978705382, −7.18427323727915640917830058966, −6.18921084075095116236138322392, −5.47194327604924545404379565562, −4.32279217912789080584434997630, −3.48418478593803961018549036198, −2.35125817870869143538302608520, −1.67444121606388394473781174539, 1.67444121606388394473781174539, 2.35125817870869143538302608520, 3.48418478593803961018549036198, 4.32279217912789080584434997630, 5.47194327604924545404379565562, 6.18921084075095116236138322392, 7.18427323727915640917830058966, 7.85564758843381750571978705382, 8.780910300492600097267113344300, 9.292538845169000671451836209939

Graph of the ZZ-function along the critical line