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Properties

Label 2-507-13.12-c3-0-6
Degree $2$
Conductor $507$
Sign $0.691 - 0.722i$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.691 - 0.722i$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ 0.691 - 0.722i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7697587876\)
\(L(\frac12)\) \(\approx\) \(0.7697587876\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
13 \( 1 \)
good2 \( 1 + 4.83iT - 8T^{2} \)
5 \( 1 - 21.1iT - 125T^{2} \)
7 \( 1 + 16.2iT - 343T^{2} \)
11 \( 1 + 30.7iT - 1.33e3T^{2} \)
17 \( 1 + 46.2T + 4.91e3T^{2} \)
19 \( 1 - 144. iT - 6.85e3T^{2} \)
23 \( 1 + 8.38T + 1.21e4T^{2} \)
29 \( 1 + 242.T + 2.43e4T^{2} \)
31 \( 1 - 87.9iT - 2.97e4T^{2} \)
37 \( 1 - 49.6iT - 5.06e4T^{2} \)
41 \( 1 + 107. iT - 6.89e4T^{2} \)
43 \( 1 - 35.4T + 7.95e4T^{2} \)
47 \( 1 - 374. iT - 1.03e5T^{2} \)
53 \( 1 + 348.T + 1.48e5T^{2} \)
59 \( 1 - 679. iT - 2.05e5T^{2} \)
61 \( 1 + 230.T + 2.26e5T^{2} \)
67 \( 1 - 295. iT - 3.00e5T^{2} \)
71 \( 1 + 329. iT - 3.57e5T^{2} \)
73 \( 1 - 48.9iT - 3.89e5T^{2} \)
79 \( 1 + 107.T + 4.93e5T^{2} \)
83 \( 1 + 515. iT - 5.71e5T^{2} \)
89 \( 1 - 984. iT - 7.04e5T^{2} \)
97 \( 1 + 487. iT - 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line