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Properties

Label 2-531-59.58-c4-0-65
Degree $2$
Conductor $531$
Sign $0.960 + 0.277i$
Analytic cond. $54.8894$
Root an. cond. $7.40874$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.15i·2-s + 11.3·4-s − 30.9·5-s + 73.3·7-s + 58.9i·8-s − 66.6i·10-s − 203. i·11-s + 115. i·13-s + 158. i·14-s + 54.5·16-s + 149.·17-s − 97.5·19-s − 351.·20-s + 438.·22-s − 785. i·23-s + ⋯
L(s)  = 1  + 0.538i·2-s + 0.709·4-s − 1.23·5-s + 1.49·7-s + 0.921i·8-s − 0.666i·10-s − 1.68i·11-s + 0.681i·13-s + 0.806i·14-s + 0.213·16-s + 0.517·17-s − 0.270·19-s − 0.878·20-s + 0.906·22-s − 1.48i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.960 + 0.277i$
Analytic conductor: \(54.8894\)
Root analytic conductor: \(7.40874\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :2),\ 0.960 + 0.277i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.295033880\)
\(L(\frac12)\) \(\approx\) \(2.295033880\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 + (3.34e3 + 964. i)T \)
good2 \( 1 - 2.15iT - 16T^{2} \)
5 \( 1 + 30.9T + 625T^{2} \)
7 \( 1 - 73.3T + 2.40e3T^{2} \)
11 \( 1 + 203. iT - 1.46e4T^{2} \)
13 \( 1 - 115. iT - 2.85e4T^{2} \)
17 \( 1 - 149.T + 8.35e4T^{2} \)
19 \( 1 + 97.5T + 1.30e5T^{2} \)
23 \( 1 + 785. iT - 2.79e5T^{2} \)
29 \( 1 + 279.T + 7.07e5T^{2} \)
31 \( 1 + 520. iT - 9.23e5T^{2} \)
37 \( 1 + 737. iT - 1.87e6T^{2} \)
41 \( 1 + 540.T + 2.82e6T^{2} \)
43 \( 1 + 1.17e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.50e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.63e3T + 7.89e6T^{2} \)
61 \( 1 + 5.34e3iT - 1.38e7T^{2} \)
67 \( 1 + 1.69e3iT - 2.01e7T^{2} \)
71 \( 1 - 9.06e3T + 2.54e7T^{2} \)
73 \( 1 - 876. iT - 2.83e7T^{2} \)
79 \( 1 - 7.89e3T + 3.89e7T^{2} \)
83 \( 1 - 3.53e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.14e4iT - 6.27e7T^{2} \)
97 \( 1 + 1.51e4iT - 8.85e7T^{2} \)
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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

−10.65149886439106021082418792626, −8.816898037918575860665467876897, −8.156687393717238292008210413151, −7.71811719758424524079715150158, −6.63769523946134443888880509401, −5.63752182008282626456701694398, −4.57934265155524409952720563847, −3.48522627305143533325073655237, −2.09762622900116108416973577820, −0.63984056594571631474794570034, 1.16139616474366652450633116961, 2.10462444601589398919402014083, 3.48013571930391607540563206154, 4.42383930254365471146793722652, 5.40323946511880730757742189472, 7.03487796935482620504076913517, 7.60556479082490234606526769999, 8.197274039085538593410218379357, 9.632153722560381914986846224720, 10.53405884096909846596336248247

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