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Properties

Label 2-507-1.1-c3-0-24
Degree 22
Conductor 507507
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.17·2-s + 3·3-s + 2.07·4-s + 6.74·5-s − 9.52·6-s − 14.1·7-s + 18.8·8-s + 9·9-s − 21.3·10-s + 62.4·11-s + 6.21·12-s + 44.9·14-s + 20.2·15-s − 76.2·16-s − 58.6·17-s − 28.5·18-s + 64.1·19-s + 13.9·20-s − 42.5·21-s − 198.·22-s + 10.9·23-s + 56.4·24-s − 79.5·25-s + 27·27-s − 29.3·28-s + 216.·29-s − 64.1·30-s + ⋯
L(s)  = 1  − 1.12·2-s + 0.577·3-s + 0.258·4-s + 0.602·5-s − 0.647·6-s − 0.765·7-s + 0.831·8-s + 0.333·9-s − 0.676·10-s + 1.71·11-s + 0.149·12-s + 0.858·14-s + 0.348·15-s − 1.19·16-s − 0.836·17-s − 0.373·18-s + 0.774·19-s + 0.156·20-s − 0.441·21-s − 1.92·22-s + 0.0990·23-s + 0.480·24-s − 0.636·25-s + 0.192·27-s − 0.198·28-s + 1.38·29-s − 0.390·30-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: 11
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: 00
Selberg data: (2, 507, ( :3/2), 1)(2,\ 507,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.4123721171.412372117
L(12)L(\frac12) ≈\approx 1.4123721171.412372117
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+3.17T+8T2 1 + 3.17T + 8T^{2}
5 1−6.74T+125T2 1 - 6.74T + 125T^{2}
7 1+14.1T+343T2 1 + 14.1T + 343T^{2}
11 1−62.4T+1.33e3T2 1 - 62.4T + 1.33e3T^{2}
17 1+58.6T+4.91e3T2 1 + 58.6T + 4.91e3T^{2}
19 1−64.1T+6.85e3T2 1 - 64.1T + 6.85e3T^{2}
23 1−10.9T+1.21e4T2 1 - 10.9T + 1.21e4T^{2}
29 1−216.T+2.43e4T2 1 - 216.T + 2.43e4T^{2}
31 1+38.6T+2.97e4T2 1 + 38.6T + 2.97e4T^{2}
37 1−423.T+5.06e4T2 1 - 423.T + 5.06e4T^{2}
41 1+366.T+6.89e4T2 1 + 366.T + 6.89e4T^{2}
43 1+128.T+7.95e4T2 1 + 128.T + 7.95e4T^{2}
47 1+93.1T+1.03e5T2 1 + 93.1T + 1.03e5T^{2}
53 1−131.T+1.48e5T2 1 - 131.T + 1.48e5T^{2}
59 1−386.T+2.05e5T2 1 - 386.T + 2.05e5T^{2}
61 1+621.T+2.26e5T2 1 + 621.T + 2.26e5T^{2}
67 1−865.T+3.00e5T2 1 - 865.T + 3.00e5T^{2}
71 1+607.T+3.57e5T2 1 + 607.T + 3.57e5T^{2}
73 1−980.T+3.89e5T2 1 - 980.T + 3.89e5T^{2}
79 1−1.33e3T+4.93e5T2 1 - 1.33e3T + 4.93e5T^{2}
83 1−907.T+5.71e5T2 1 - 907.T + 5.71e5T^{2}
89 1+1.03e3T+7.04e5T2 1 + 1.03e3T + 7.04e5T^{2}
97 1−1.04e3T+9.12e5T2 1 - 1.04e3T + 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

−9.937782183640601512661676818711, −9.582566758672875821991852670081, −8.938736073904058880209565818430, −8.095628366696897589884875362340, −6.92011381290682185407695689889, −6.29939397454874708465557563107, −4.64954552111971146038011981969, −3.51405929528235664655369012913, −2.03731445457537142740748997824, −0.886995777907303732887522010920, 0.886995777907303732887522010920, 2.03731445457537142740748997824, 3.51405929528235664655369012913, 4.64954552111971146038011981969, 6.29939397454874708465557563107, 6.92011381290682185407695689889, 8.095628366696897589884875362340, 8.938736073904058880209565818430, 9.582566758672875821991852670081, 9.937782183640601512661676818711

Graph of the ZZ-function along the critical line