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Properties

Label 2-531-3.2-c4-0-13
Degree $2$
Conductor $531$
Sign $0.816 + 0.577i$
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  − 6.05i·2-s − 20.6·4-s − 34.3i·5-s − 5.09·7-s + 28.1i·8-s − 207.·10-s + 148. i·11-s + 69.3·13-s + 30.8i·14-s − 160.·16-s + 321. i·17-s + 76.0·19-s + 708. i·20-s + 899.·22-s + 232. i·23-s + ⋯
L(s)  = 1  − 1.51i·2-s − 1.29·4-s − 1.37i·5-s − 0.104·7-s + 0.439i·8-s − 2.07·10-s + 1.22i·11-s + 0.410·13-s + 0.157i·14-s − 0.625·16-s + 1.11i·17-s + 0.210·19-s + 1.77i·20-s + 1.85·22-s + 0.439i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(54.8894\)
Root analytic conductor: \(7.40874\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :2),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.161073648\)
\(L(\frac12)\) \(\approx\) \(1.161073648\)
\(L(3)\) 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 \( 1 - 453. iT \)
good2 \( 1 + 6.05iT - 16T^{2} \)
5 \( 1 + 34.3iT - 625T^{2} \)
7 \( 1 + 5.09T + 2.40e3T^{2} \)
11 \( 1 - 148. iT - 1.46e4T^{2} \)
13 \( 1 - 69.3T + 2.85e4T^{2} \)
17 \( 1 - 321. iT - 8.35e4T^{2} \)
19 \( 1 - 76.0T + 1.30e5T^{2} \)
23 \( 1 - 232. iT - 2.79e5T^{2} \)
29 \( 1 - 806. iT - 7.07e5T^{2} \)
31 \( 1 - 882.T + 9.23e5T^{2} \)
37 \( 1 + 1.27e3T + 1.87e6T^{2} \)
41 \( 1 - 2.40e3iT - 2.82e6T^{2} \)
43 \( 1 - 358.T + 3.41e6T^{2} \)
47 \( 1 - 1.99e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.99e3iT - 7.89e6T^{2} \)
61 \( 1 + 6.44e3T + 1.38e7T^{2} \)
67 \( 1 + 2.78e3T + 2.01e7T^{2} \)
71 \( 1 + 6.19e3iT - 2.54e7T^{2} \)
73 \( 1 - 5.12e3T + 2.83e7T^{2} \)
79 \( 1 - 4.18e3T + 3.89e7T^{2} \)
83 \( 1 - 3.30e3iT - 4.74e7T^{2} \)
89 \( 1 + 4.94e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.62e4T + 8.85e7T^{2} \)
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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

−10.20402099231522845157190363379, −9.436461027709922362797969137061, −8.788990834882759237726623441626, −7.76646496658360536669957341021, −6.35649361099730610007469524360, −4.94302620379789222585708864514, −4.34312749192583256155891814567, −3.21974333141010757897802489238, −1.76260849442844614817838254078, −1.17324784293204687899133107414, 0.32092517589255072729445490087, 2.57442959950478508235611946156, 3.62913811137176210261807562083, 5.07243473429112366335852108486, 6.07373379378301703464386865368, 6.63413097543601450514965856670, 7.42505939846137082589683141499, 8.277057501484316230894119695869, 9.139821344700990583595924044585, 10.28256024077310991488533128308

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