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Properties

Label 2-104-104.51-c0-0-0
Degree 22
Conductor 104104
Sign 11
Analytic cond. 0.05190270.0519027
Root an. cond. 0.2278210.227821
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  − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s − 10-s − 12-s − 13-s − 14-s − 15-s + 16-s − 17-s + 20-s − 21-s + 24-s + 26-s + 27-s + 28-s + 30-s − 2·31-s − 32-s + 34-s + 35-s + 37-s + 39-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s − 10-s − 12-s − 13-s − 14-s − 15-s + 16-s − 17-s + 20-s − 21-s + 24-s + 26-s + 27-s + 28-s + 30-s − 2·31-s − 32-s + 34-s + 35-s + 37-s + 39-s + ⋯

Functional equation

Λ(s)=(104s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(104s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 104104    =    23⋅132^{3} \cdot 13
Sign: 11
Analytic conductor: 0.05190270.0519027
Root analytic conductor: 0.2278210.227821
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ104(51,⋅)\chi_{104} (51, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 104, ( :0), 1)(2,\ 104,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.35064456210.3506445621
L(12)L(\frac12) ≈\approx 0.35064456210.3506445621
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
13 1+T 1 + T
good3 1+T+T2 1 + T + T^{2}
5 1−T+T2 1 - T + T^{2}
7 1−T+T2 1 - T + T^{2}
11 (1−T)(1+T) ( 1 - T )( 1 + T )
17 1+T+T2 1 + T + T^{2}
19 (1−T)(1+T) ( 1 - T )( 1 + T )
23 (1−T)(1+T) ( 1 - T )( 1 + T )
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 (1+T)2 ( 1 + T )^{2}
37 1−T+T2 1 - T + T^{2}
41 (1−T)(1+T) ( 1 - T )( 1 + T )
43 1+T+T2 1 + T + T^{2}
47 1−T+T2 1 - T + T^{2}
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 (1−T)(1+T) ( 1 - T )( 1 + T )
61 (1−T)(1+T) ( 1 - T )( 1 + T )
67 (1−T)(1+T) ( 1 - T )( 1 + T )
71 1−T+T2 1 - T + T^{2}
73 (1−T)(1+T) ( 1 - T )( 1 + T )
79 (1−T)(1+T) ( 1 - T )( 1 + T )
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

−14.21903403686422288216786219204, −12.69759417877772506687250298073, −11.53875891143008359430010850022, −10.91362787156601425843494571816, −9.849520919914033047471247341525, −8.768893274904782034971084298372, −7.37748796465981917417358202634, −6.13973319085271586429221786246, −5.11000336544565001207901214497, −2.09431808749519266596519118860, 2.09431808749519266596519118860, 5.11000336544565001207901214497, 6.13973319085271586429221786246, 7.37748796465981917417358202634, 8.768893274904782034971084298372, 9.849520919914033047471247341525, 10.91362787156601425843494571816, 11.53875891143008359430010850022, 12.69759417877772506687250298073, 14.21903403686422288216786219204

Graph of the ZZ-function along the critical line