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Properties

Label 2-507-1.1-c3-0-2
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  − 2.04·2-s + 3·3-s − 3.82·4-s − 12.0·5-s − 6.12·6-s − 29.7·7-s + 24.1·8-s + 9·9-s + 24.6·10-s − 28.0·11-s − 11.4·12-s + 60.7·14-s − 36.2·15-s − 18.7·16-s − 50.6·17-s − 18.3·18-s − 105.·19-s + 46.2·20-s − 89.2·21-s + 57.3·22-s − 160.·23-s + 72.4·24-s + 20.9·25-s + 27·27-s + 113.·28-s + 140.·29-s + 74.0·30-s + ⋯
L(s)  = 1  − 0.722·2-s + 0.577·3-s − 0.478·4-s − 1.08·5-s − 0.416·6-s − 1.60·7-s + 1.06·8-s + 0.333·9-s + 0.780·10-s − 0.769·11-s − 0.276·12-s + 1.15·14-s − 0.623·15-s − 0.292·16-s − 0.722·17-s − 0.240·18-s − 1.26·19-s + 0.517·20-s − 0.927·21-s + 0.555·22-s − 1.45·23-s + 0.616·24-s + 0.167·25-s + 0.192·27-s + 0.768·28-s + 0.897·29-s + 0.450·30-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 0.28375347620.2837534762
L(12)L(\frac12) ≈\approx 0.28375347620.2837534762
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+2.04T+8T2 1 + 2.04T + 8T^{2}
5 1+12.0T+125T2 1 + 12.0T + 125T^{2}
7 1+29.7T+343T2 1 + 29.7T + 343T^{2}
11 1+28.0T+1.33e3T2 1 + 28.0T + 1.33e3T^{2}
17 1+50.6T+4.91e3T2 1 + 50.6T + 4.91e3T^{2}
19 1+105.T+6.85e3T2 1 + 105.T + 6.85e3T^{2}
23 1+160.T+1.21e4T2 1 + 160.T + 1.21e4T^{2}
29 1−140.T+2.43e4T2 1 - 140.T + 2.43e4T^{2}
31 1+223.T+2.97e4T2 1 + 223.T + 2.97e4T^{2}
37 1−228.T+5.06e4T2 1 - 228.T + 5.06e4T^{2}
41 1−295.T+6.89e4T2 1 - 295.T + 6.89e4T^{2}
43 1−192.T+7.95e4T2 1 - 192.T + 7.95e4T^{2}
47 1+36.9T+1.03e5T2 1 + 36.9T + 1.03e5T^{2}
53 1−149.T+1.48e5T2 1 - 149.T + 1.48e5T^{2}
59 1+438.T+2.05e5T2 1 + 438.T + 2.05e5T^{2}
61 1−286.T+2.26e5T2 1 - 286.T + 2.26e5T^{2}
67 1−537.T+3.00e5T2 1 - 537.T + 3.00e5T^{2}
71 1+102.T+3.57e5T2 1 + 102.T + 3.57e5T^{2}
73 1−75.5T+3.89e5T2 1 - 75.5T + 3.89e5T^{2}
79 1−17.5T+4.93e5T2 1 - 17.5T + 4.93e5T^{2}
83 1+1.46e3T+5.71e5T2 1 + 1.46e3T + 5.71e5T^{2}
89 1−334.T+7.04e5T2 1 - 334.T + 7.04e5T^{2}
97 1−748.T+9.12e5T2 1 - 748.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

−10.24781102541153325182718732055, −9.537176155319608814046380898982, −8.700209915987861492523847717149, −7.984635170979513700630639701958, −7.20195118050387779115268511079, −6.06315207404290797158305446601, −4.38195524046886459599132704104, −3.74783416555354025034339180164, −2.41995946035335524971548540966, −0.33755533381617029524801544648, 0.33755533381617029524801544648, 2.41995946035335524971548540966, 3.74783416555354025034339180164, 4.38195524046886459599132704104, 6.06315207404290797158305446601, 7.20195118050387779115268511079, 7.984635170979513700630639701958, 8.700209915987861492523847717149, 9.537176155319608814046380898982, 10.24781102541153325182718732055

Graph of the ZZ-function along the critical line