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Properties

Label 2-507-13.12-c3-0-6
Degree 22
Conductor 507507
Sign 0.691−0.722i0.691 - 0.722i
Analytic cond. 29.913929.9139
Root an. cond. 5.469365.46936
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.83i·2-s + 3·3-s − 15.4·4-s + 21.1i·5-s − 14.5i·6-s − 16.2i·7-s + 35.8i·8-s + 9·9-s + 102.·10-s − 30.7i·11-s − 46.2·12-s − 78.7·14-s + 63.5i·15-s + 49.9·16-s − 46.2·17-s − 43.5i·18-s + ⋯
L(s)  = 1  − 1.71i·2-s + 0.577·3-s − 1.92·4-s + 1.89i·5-s − 0.987i·6-s − 0.879i·7-s + 1.58i·8-s + 0.333·9-s + 3.24·10-s − 0.842i·11-s − 1.11·12-s − 1.50·14-s + 1.09i·15-s + 0.781·16-s − 0.659·17-s − 0.570i·18-s + ⋯

Functional equation

Λ(s)=(507s/2 ΓC(s) L(s)=((0.691−0.722i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(507s/2 ΓC(s+3/2) L(s)=((0.691−0.722i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 507507    =    3⋅1323 \cdot 13^{2}
Sign: 0.691−0.722i0.691 - 0.722i
Analytic conductor: 29.913929.9139
Root analytic conductor: 5.469365.46936
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ507(337,⋅)\chi_{507} (337, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 507, ( :3/2), 0.691−0.722i)(2,\ 507,\ (\ :3/2),\ 0.691 - 0.722i)

Particular Values

L(2)L(2) ≈\approx 0.76975878760.7697587876
L(12)L(\frac12) ≈\approx 0.76975878760.7697587876
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+4.83iT−8T2 1 + 4.83iT - 8T^{2}
5 1−21.1iT−125T2 1 - 21.1iT - 125T^{2}
7 1+16.2iT−343T2 1 + 16.2iT - 343T^{2}
11 1+30.7iT−1.33e3T2 1 + 30.7iT - 1.33e3T^{2}
17 1+46.2T+4.91e3T2 1 + 46.2T + 4.91e3T^{2}
19 1−144.iT−6.85e3T2 1 - 144. iT - 6.85e3T^{2}
23 1+8.38T+1.21e4T2 1 + 8.38T + 1.21e4T^{2}
29 1+242.T+2.43e4T2 1 + 242.T + 2.43e4T^{2}
31 1−87.9iT−2.97e4T2 1 - 87.9iT - 2.97e4T^{2}
37 1−49.6iT−5.06e4T2 1 - 49.6iT - 5.06e4T^{2}
41 1+107.iT−6.89e4T2 1 + 107. iT - 6.89e4T^{2}
43 1−35.4T+7.95e4T2 1 - 35.4T + 7.95e4T^{2}
47 1−374.iT−1.03e5T2 1 - 374. iT - 1.03e5T^{2}
53 1+348.T+1.48e5T2 1 + 348.T + 1.48e5T^{2}
59 1−679.iT−2.05e5T2 1 - 679. iT - 2.05e5T^{2}
61 1+230.T+2.26e5T2 1 + 230.T + 2.26e5T^{2}
67 1−295.iT−3.00e5T2 1 - 295. iT - 3.00e5T^{2}
71 1+329.iT−3.57e5T2 1 + 329. iT - 3.57e5T^{2}
73 1−48.9iT−3.89e5T2 1 - 48.9iT - 3.89e5T^{2}
79 1+107.T+4.93e5T2 1 + 107.T + 4.93e5T^{2}
83 1+515.iT−5.71e5T2 1 + 515. iT - 5.71e5T^{2}
89 1−984.iT−7.04e5T2 1 - 984. iT - 7.04e5T^{2}
97 1+487.iT−9.12e5T2 1 + 487. iT - 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

−10.70254443381968895390093615314, −10.13966927119719435330797694597, −9.278336517291027318023402754129, −8.025969065700759238334236012205, −7.14001892148921336226860245051, −5.98115367317070440320982540806, −4.05045527924138206545689739424, −3.52200562011416810422488841754, −2.69857159354379117953208204264, −1.58764120664894426844451116800, 0.21301278575045388790690397303, 2.03969023057303742020992110103, 4.22305865582791278384818715106, 4.92776255175663037163681101670, 5.60566297414986296355898668770, 6.81585644267410678857273426985, 7.79554465618311273911447652719, 8.520391936802824005448954779052, 9.281272496264218589208539930072, 9.408026157993062936895142051829

Graph of the ZZ-function along the critical line