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Properties

Label 2-507-13.12-c3-0-52
Degree $2$
Conductor $507$
Sign $-0.246 + 0.969i$
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  − 3.17i·2-s + 3·3-s − 2.07·4-s + 6.74i·5-s − 9.52i·6-s + 14.1i·7-s − 18.8i·8-s + 9·9-s + 21.3·10-s − 62.4i·11-s − 6.21·12-s + 44.9·14-s + 20.2i·15-s − 76.2·16-s + 58.6·17-s − 28.5i·18-s + ⋯
L(s)  = 1  − 1.12i·2-s + 0.577·3-s − 0.258·4-s + 0.602i·5-s − 0.647i·6-s + 0.765i·7-s − 0.831i·8-s + 0.333·9-s + 0.676·10-s − 1.71i·11-s − 0.149·12-s + 0.858·14-s + 0.348i·15-s − 1.19·16-s + 0.836·17-s − 0.373i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.246 + 0.969i$
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: \((2,\ 507,\ (\ :3/2),\ -0.246 + 0.969i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.690186513\)
\(L(\frac12)\) \(\approx\) \(2.690186513\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.17iT - 8T^{2} \)
5 \( 1 - 6.74iT - 125T^{2} \)
7 \( 1 - 14.1iT - 343T^{2} \)
11 \( 1 + 62.4iT - 1.33e3T^{2} \)
17 \( 1 - 58.6T + 4.91e3T^{2} \)
19 \( 1 - 64.1iT - 6.85e3T^{2} \)
23 \( 1 + 10.9T + 1.21e4T^{2} \)
29 \( 1 - 216.T + 2.43e4T^{2} \)
31 \( 1 + 38.6iT - 2.97e4T^{2} \)
37 \( 1 + 423. iT - 5.06e4T^{2} \)
41 \( 1 + 366. iT - 6.89e4T^{2} \)
43 \( 1 - 128.T + 7.95e4T^{2} \)
47 \( 1 - 93.1iT - 1.03e5T^{2} \)
53 \( 1 - 131.T + 1.48e5T^{2} \)
59 \( 1 + 386. iT - 2.05e5T^{2} \)
61 \( 1 + 621.T + 2.26e5T^{2} \)
67 \( 1 - 865. iT - 3.00e5T^{2} \)
71 \( 1 + 607. iT - 3.57e5T^{2} \)
73 \( 1 + 980. iT - 3.89e5T^{2} \)
79 \( 1 - 1.33e3T + 4.93e5T^{2} \)
83 \( 1 - 907. iT - 5.71e5T^{2} \)
89 \( 1 - 1.03e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.04e3iT - 9.12e5T^{2} \)
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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

−10.59607328032801562033204228184, −9.445543585888543024122923337807, −8.696571389396061351755341067801, −7.71333062814265833637443828321, −6.49303883010968486698091597278, −5.57957774288642154794268697604, −3.83168470774886893913298732614, −3.10370482718583795793431989372, −2.30032267758586692883884011105, −0.855319683543282314363401569872, 1.31670847039631941896428621874, 2.78097172705998923130421181817, 4.48019341198564027416073325647, 4.96772110636489456950279776149, 6.47169537183475918273584962272, 7.17984647383419000522928376580, 7.87926090488436130569503335152, 8.706734262054053400446822217791, 9.714116051805101991242311098068, 10.46867143235028016155075332924

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