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Properties

Label 2-504-56.13-c0-0-2
Degree 22
Conductor 504504
Sign 11
Analytic cond. 0.2515280.251528
Root an. cond. 0.5015260.501526
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 + 4-s − 7-s + 8-s − 14-s + 16-s − 2·23-s − 25-s − 28-s + 32-s − 2·46-s + 49-s − 50-s − 56-s + 64-s + 2·71-s − 2·79-s − 2·92-s + 98-s − 100-s − 112-s + 2·113-s + ⋯
L(s)  = 1  + 2-s + 4-s − 7-s + 8-s − 14-s + 16-s − 2·23-s − 25-s − 28-s + 32-s − 2·46-s + 49-s − 50-s − 56-s + 64-s + 2·71-s − 2·79-s − 2·92-s + 98-s − 100-s − 112-s + 2·113-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: 11
Analytic conductor: 0.2515280.251528
Root analytic conductor: 0.5015260.501526
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ504(181,⋅)\chi_{504} (181, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 504, ( :0), 1)(2,\ 504,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4398931001.439893100
L(12)L(\frac12) ≈\approx 1.4398931001.439893100
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
3 1 1
7 1+T 1 + T
good5 1+T2 1 + T^{2}
11 (1−T)(1+T) ( 1 - T )( 1 + T )
13 1+T2 1 + T^{2}
17 (1−T)(1+T) ( 1 - T )( 1 + T )
19 1+T2 1 + T^{2}
23 (1+T)2 ( 1 + T )^{2}
29 (1−T)(1+T) ( 1 - T )( 1 + T )
31 (1−T)(1+T) ( 1 - T )( 1 + T )
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)(1+T) ( 1 - T )( 1 + T )
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 1+T2 1 + T^{2}
61 1+T2 1 + T^{2}
67 (1−T)(1+T) ( 1 - T )( 1 + T )
71 (1−T)2 ( 1 - T )^{2}
73 (1−T)(1+T) ( 1 - T )( 1 + T )
79 (1+T)2 ( 1 + T )^{2}
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

−11.36883216382947901268222888836, −10.26281683024352818903610460981, −9.666087849294455038237157313933, −8.255353598993736633003017504851, −7.28424641630970624002389000280, −6.29115887013593533422292284790, −5.65539355150237745385542105144, −4.28631963116974234825053477736, −3.43363451204020501313795054491, −2.15848280631622720996979725155, 2.15848280631622720996979725155, 3.43363451204020501313795054491, 4.28631963116974234825053477736, 5.65539355150237745385542105144, 6.29115887013593533422292284790, 7.28424641630970624002389000280, 8.255353598993736633003017504851, 9.666087849294455038237157313933, 10.26281683024352818903610460981, 11.36883216382947901268222888836

Graph of the ZZ-function along the critical line