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Properties

Label 2-507-1.1-c3-0-52
Degree 22
Conductor 507507
Sign −1-1
Analytic cond. 29.913929.9139
Root an. cond. 5.469365.46936
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.52·2-s + 3·3-s + 22.4·4-s + 6.08·5-s − 16.5·6-s − 20.2·7-s − 80.0·8-s + 9·9-s − 33.5·10-s + 48.8·11-s + 67.4·12-s + 111.·14-s + 18.2·15-s + 262.·16-s − 37.7·17-s − 49.7·18-s − 120.·19-s + 136.·20-s − 60.8·21-s − 269.·22-s + 74.8·23-s − 240.·24-s − 88.0·25-s + 27·27-s − 456.·28-s − 112.·29-s − 100.·30-s + ⋯
L(s)  = 1  − 1.95·2-s + 0.577·3-s + 2.81·4-s + 0.543·5-s − 1.12·6-s − 1.09·7-s − 3.53·8-s + 0.333·9-s − 1.06·10-s + 1.33·11-s + 1.62·12-s + 2.13·14-s + 0.314·15-s + 4.09·16-s − 0.538·17-s − 0.650·18-s − 1.45·19-s + 1.52·20-s − 0.631·21-s − 2.61·22-s + 0.678·23-s − 2.04·24-s − 0.704·25-s + 0.192·27-s − 3.07·28-s − 0.721·29-s − 0.613·30-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 29.913929.9139
Root analytic conductor: 5.469365.46936
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 507, ( :3/2), −1)(2,\ 507,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) 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 1−3T 1 - 3T
13 1 1
good2 1+5.52T+8T2 1 + 5.52T + 8T^{2}
5 1−6.08T+125T2 1 - 6.08T + 125T^{2}
7 1+20.2T+343T2 1 + 20.2T + 343T^{2}
11 1−48.8T+1.33e3T2 1 - 48.8T + 1.33e3T^{2}
17 1+37.7T+4.91e3T2 1 + 37.7T + 4.91e3T^{2}
19 1+120.T+6.85e3T2 1 + 120.T + 6.85e3T^{2}
23 1−74.8T+1.21e4T2 1 - 74.8T + 1.21e4T^{2}
29 1+112.T+2.43e4T2 1 + 112.T + 2.43e4T^{2}
31 1+113.T+2.97e4T2 1 + 113.T + 2.97e4T^{2}
37 1−85.7T+5.06e4T2 1 - 85.7T + 5.06e4T^{2}
41 1−133.T+6.89e4T2 1 - 133.T + 6.89e4T^{2}
43 1+319.T+7.95e4T2 1 + 319.T + 7.95e4T^{2}
47 1−401.T+1.03e5T2 1 - 401.T + 1.03e5T^{2}
53 1+384.T+1.48e5T2 1 + 384.T + 1.48e5T^{2}
59 1−121.T+2.05e5T2 1 - 121.T + 2.05e5T^{2}
61 1−220.T+2.26e5T2 1 - 220.T + 2.26e5T^{2}
67 1+975.T+3.00e5T2 1 + 975.T + 3.00e5T^{2}
71 1+106.T+3.57e5T2 1 + 106.T + 3.57e5T^{2}
73 1+43.2T+3.89e5T2 1 + 43.2T + 3.89e5T^{2}
79 1−539.T+4.93e5T2 1 - 539.T + 4.93e5T^{2}
83 1+811.T+5.71e5T2 1 + 811.T + 5.71e5T^{2}
89 1−1.13e3T+7.04e5T2 1 - 1.13e3T + 7.04e5T^{2}
97 1−229.T+9.12e5T2 1 - 229.T + 9.12e5T^{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

−9.716125657346578332865629553293, −9.167301524070757441263745039564, −8.681393894204845183816660822259, −7.48686275777391827635344128964, −6.61077475051217236407874608553, −6.11326066466291660911906801148, −3.72881508368956882778015543095, −2.49189862732150881736520245221, −1.51361866498344786989979103022, 0, 1.51361866498344786989979103022, 2.49189862732150881736520245221, 3.72881508368956882778015543095, 6.11326066466291660911906801148, 6.61077475051217236407874608553, 7.48686275777391827635344128964, 8.681393894204845183816660822259, 9.167301524070757441263745039564, 9.716125657346578332865629553293

Graph of the ZZ-function along the critical line