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Properties

Label 2-496-31.30-c0-0-0
Degree 22
Conductor 496496
Sign 11
Analytic cond. 0.2475360.247536
Root an. cond. 0.4975300.497530
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  − 5-s + 7-s + 9-s + 19-s − 31-s − 35-s − 41-s − 45-s − 2·47-s + 59-s + 63-s − 2·67-s + 71-s + 81-s − 95-s − 97-s − 101-s + 103-s + 107-s − 109-s − 113-s + ⋯
L(s)  = 1  − 5-s + 7-s + 9-s + 19-s − 31-s − 35-s − 41-s − 45-s − 2·47-s + 59-s + 63-s − 2·67-s + 71-s + 81-s − 95-s − 97-s − 101-s + 103-s + 107-s − 109-s − 113-s + ⋯

Functional equation

Λ(s)=(496s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(496s/2 ΓC(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 496496    =    24⋅312^{4} \cdot 31
Sign: 11
Analytic conductor: 0.2475360.247536
Root analytic conductor: 0.4975300.497530
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ496(433,⋅)\chi_{496} (433, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 496, ( :0), 1)(2,\ 496,\ (\ :0),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.87396577990.8739657799
L(12)L(\frac12) ≈\approx 0.87396577990.8739657799
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
31 1+T 1 + T
good3 (1−T)(1+T) ( 1 - T )( 1 + T )
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 )
13 (1−T)(1+T) ( 1 - T )( 1 + T )
17 (1−T)(1+T) ( 1 - T )( 1 + T )
19 1−T+T2 1 - T + T^{2}
23 (1−T)(1+T) ( 1 - T )( 1 + T )
29 (1−T)(1+T) ( 1 - T )( 1 + T )
37 (1−T)(1+T) ( 1 - T )( 1 + T )
41 1+T+T2 1 + T + T^{2}
43 (1−T)(1+T) ( 1 - T )( 1 + T )
47 (1+T)2 ( 1 + T )^{2}
53 (1−T)(1+T) ( 1 - T )( 1 + T )
59 1−T+T2 1 - T + T^{2}
61 (1−T)(1+T) ( 1 - T )( 1 + T )
67 (1+T)2 ( 1 + T )^{2}
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+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

−11.33979621220307494142617341998, −10.35092728363533814002009073535, −9.400971048775264223122950405864, −8.236781727676124108403907950421, −7.64178938780781159262121842857, −6.82862860558713448622581211010, −5.30668591867096172288592059861, −4.43474446882544503532194136997, −3.44767844354801433253755389737, −1.61544784124339313944370287833, 1.61544784124339313944370287833, 3.44767844354801433253755389737, 4.43474446882544503532194136997, 5.30668591867096172288592059861, 6.82862860558713448622581211010, 7.64178938780781159262121842857, 8.236781727676124108403907950421, 9.400971048775264223122950405864, 10.35092728363533814002009073535, 11.33979621220307494142617341998

Graph of the ZZ-function along the critical line