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Properties

Label 2-728-728.181-c0-0-2
Degree 22
Conductor 728728
Sign 11
Analytic cond. 0.3633190.363319
Root an. cond. 0.6027590.602759
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 − 6-s + 7-s − 8-s + 11-s + 12-s − 13-s − 14-s + 16-s + 21-s − 22-s − 23-s − 24-s + 25-s + 26-s − 27-s + 28-s − 31-s − 32-s + 33-s + 37-s − 39-s − 41-s − 42-s + 44-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 11-s + 12-s − 13-s − 14-s + 16-s + 21-s − 22-s − 23-s − 24-s + 25-s + 26-s − 27-s + 28-s − 31-s − 32-s + 33-s + 37-s − 39-s − 41-s − 42-s + 44-s + ⋯

Functional equation

Λ(s)=(728s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(728s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 728728    =    23⋅7⋅132^{3} \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 0.3633190.363319
Root analytic conductor: 0.6027590.602759
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ728(181,⋅)\chi_{728} (181, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 728, ( :0), 1)(2,\ 728,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.91579757620.9157975762
L(12)L(\frac12) ≈\approx 0.91579757620.9157975762
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
7 1−T 1 - T
13 1+T 1 + T
good3 1−T+T2 1 - T + T^{2}
5 (1−T)(1+T) ( 1 - T )( 1 + T )
11 1−T+T2 1 - T + T^{2}
17 (1−T)(1+T) ( 1 - T )( 1 + T )
19 (1−T)(1+T) ( 1 - T )( 1 + T )
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+T2 1 - T + T^{2}
41 1+T+T2 1 + T + T^{2}
43 (1−T)(1+T) ( 1 - T )( 1 + T )
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+T2 1 - T + T^{2}
67 1−T+T2 1 - T + T^{2}
71 (1−T)(1+T) ( 1 - T )( 1 + T )
73 1+T+T2 1 + T + T^{2}
79 1+T+T2 1 + T + T^{2}
83 (1−T)(1+T) ( 1 - T )( 1 + T )
89 (1−T)2 ( 1 - T )^{2}
97 1+T+T2 1 + T + T^{2}
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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

−10.37921869441011245738889142431, −9.511010005843341443851104686247, −8.868716737355728499862369022957, −8.148404660797174007245273401215, −7.49922333010491836577656038357, −6.54218389364111010314616188831, −5.26987348899802349293227188662, −3.87882397059735457716023899844, −2.63845126578247891358973399219, −1.66567777863511083269438526980, 1.66567777863511083269438526980, 2.63845126578247891358973399219, 3.87882397059735457716023899844, 5.26987348899802349293227188662, 6.54218389364111010314616188831, 7.49922333010491836577656038357, 8.148404660797174007245273401215, 8.868716737355728499862369022957, 9.511010005843341443851104686247, 10.37921869441011245738889142431

Graph of the ZZ-function along the critical line