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Properties

Label 2-567-63.13-c0-0-0
Degree 22
Conductor 567567
Sign 0.642+0.766i0.642 + 0.766i
Analytic cond. 0.2829690.282969
Root an. cond. 0.5319490.531949
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 1.5i)2-s + (−1 + 1.73i)4-s + (0.5 + 0.866i)7-s + 1.73·8-s + (0.866 + 1.5i)11-s + (0.866 − 1.5i)14-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)22-s + (−0.5 − 0.866i)25-s − 2·28-s + 37-s + (−0.5 − 0.866i)43-s − 3.46·44-s + (−0.499 + 0.866i)49-s + (−0.866 + 1.5i)50-s + ⋯
L(s)  = 1  + (−0.866 − 1.5i)2-s + (−1 + 1.73i)4-s + (0.5 + 0.866i)7-s + 1.73·8-s + (0.866 + 1.5i)11-s + (0.866 − 1.5i)14-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)22-s + (−0.5 − 0.866i)25-s − 2·28-s + 37-s + (−0.5 − 0.866i)43-s − 3.46·44-s + (−0.499 + 0.866i)49-s + (−0.866 + 1.5i)50-s + ⋯

Functional equation

Λ(s)=(567s/2 ΓC(s) L(s)=((0.642+0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(567s/2 ΓC(s) L(s)=((0.642+0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 567567    =    34⋅73^{4} \cdot 7
Sign: 0.642+0.766i0.642 + 0.766i
Analytic conductor: 0.2829690.282969
Root analytic conductor: 0.5319490.531949
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ567(433,⋅)\chi_{567} (433, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 567, ( :0), 0.642+0.766i)(2,\ 567,\ (\ :0),\ 0.642 + 0.766i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.59771478000.5977147800
L(12)L(\frac12) ≈\approx 0.59771478000.5977147800
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)
bad3 1 1
7 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good2 1+(0.866+1.5i)T+(−0.5+0.866i)T2 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2}
5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(−0.866−1.5i)T+(−0.5+0.866i)T2 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2}
13 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1−T2 1 - T^{2}
19 1−T2 1 - T^{2}
23 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
29 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1−T+T2 1 - T + T^{2}
41 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
43 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+1.73T+T2 1 + 1.73T + T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
71 1−1.73T+T2 1 - 1.73T + T^{2}
73 1−T2 1 - T^{2}
79 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
83 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
89 1−T2 1 - T^{2}
97 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)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

−10.89214514367268513131693933435, −9.811412262285066525937874266798, −9.438557081258007918606429047084, −8.501579990467090465047751705009, −7.70792771142513918507748388220, −6.40570697609618678615671053531, −4.87070708722958744869486818323, −3.89043779693755897932940555218, −2.48775217801204080396921726262, −1.65885049347120618410655102747, 1.11999114332158697137500474025, 3.59818317954485894339883335564, 4.88457328426438198487730737810, 5.96802682080223111941735999602, 6.63770565603998512326944184495, 7.64659259947741919184880825894, 8.234661819459637973208029249727, 9.103201546277372344072320035939, 9.849801843384793769381485832274, 10.94347761971671174317014119425

Graph of the ZZ-function along the critical line