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Properties

Label 2-507-13.12-c3-0-24
Degree $2$
Conductor $507$
Sign $-0.832 - 0.554i$
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  + 4.20i·2-s + 3·3-s − 9.71·4-s − 11.4i·5-s + 12.6i·6-s + 11.2i·7-s − 7.22i·8-s + 9·9-s + 48.1·10-s − 25.8i·11-s − 29.1·12-s − 47.3·14-s − 34.2i·15-s − 47.3·16-s + 20.3·17-s + 37.8i·18-s + ⋯
L(s)  = 1  + 1.48i·2-s + 0.577·3-s − 1.21·4-s − 1.02i·5-s + 0.859i·6-s + 0.607i·7-s − 0.319i·8-s + 0.333·9-s + 1.52·10-s − 0.709i·11-s − 0.701·12-s − 0.904·14-s − 0.590i·15-s − 0.739·16-s + 0.290·17-s + 0.496i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.832 - 0.554i$
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: \((2,\ 507,\ (\ :3/2),\ -0.832 - 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.193167430\)
\(L(\frac12)\) \(\approx\) \(2.193167430\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.20iT - 8T^{2} \)
5 \( 1 + 11.4iT - 125T^{2} \)
7 \( 1 - 11.2iT - 343T^{2} \)
11 \( 1 + 25.8iT - 1.33e3T^{2} \)
17 \( 1 - 20.3T + 4.91e3T^{2} \)
19 \( 1 - 154. iT - 6.85e3T^{2} \)
23 \( 1 - 180.T + 1.21e4T^{2} \)
29 \( 1 + 20.4T + 2.43e4T^{2} \)
31 \( 1 - 266. iT - 2.97e4T^{2} \)
37 \( 1 + 115. iT - 5.06e4T^{2} \)
41 \( 1 - 391. iT - 6.89e4T^{2} \)
43 \( 1 + 151.T + 7.95e4T^{2} \)
47 \( 1 - 467. iT - 1.03e5T^{2} \)
53 \( 1 - 79.9T + 1.48e5T^{2} \)
59 \( 1 - 873. iT - 2.05e5T^{2} \)
61 \( 1 + 187.T + 2.26e5T^{2} \)
67 \( 1 + 609. iT - 3.00e5T^{2} \)
71 \( 1 - 248. iT - 3.57e5T^{2} \)
73 \( 1 + 852. iT - 3.89e5T^{2} \)
79 \( 1 + 331.T + 4.93e5T^{2} \)
83 \( 1 + 435. iT - 5.71e5T^{2} \)
89 \( 1 + 259. iT - 7.04e5T^{2} \)
97 \( 1 - 1.22e3iT - 9.12e5T^{2} \)
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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

−10.70244155316389117612649580467, −9.381633114196484552140371594102, −8.752694026893083502064455187919, −8.226673161867438483258195962594, −7.36748916011661805232279000482, −6.19189508322074369644158901970, −5.43001760772025262974028597076, −4.57477711605292325749370261752, −3.12508975310406562209129062664, −1.32372818322645039552055952077, 0.68692634393269458271129581463, 2.18256770494763433664149220365, 2.94516721626470788383566733450, 3.86578817074809624286021742219, 4.91614986969835765400570303458, 6.88981130146565073608571472987, 7.20314734145869321965664663151, 8.735405154173392224685815762088, 9.582324305169363215167503625960, 10.28260870288936491625132650526

Graph of the $Z$-function along the critical line