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Properties

Label 2-531-3.2-c4-0-29
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  − 7.18i·2-s − 35.6·4-s + 15.5i·5-s − 43.6·7-s + 140. i·8-s + 111.·10-s + 1.21i·11-s − 174.·13-s + 313. i·14-s + 442.·16-s + 372. i·17-s + 70.8·19-s − 552. i·20-s + 8.74·22-s + 65.1i·23-s + ⋯
L(s)  = 1  − 1.79i·2-s − 2.22·4-s + 0.620i·5-s − 0.890·7-s + 2.20i·8-s + 1.11·10-s + 0.0100i·11-s − 1.03·13-s + 1.59i·14-s + 1.72·16-s + 1.28i·17-s + 0.196·19-s − 1.38i·20-s + 0.0180·22-s + 0.123i·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

\(L(\frac{5}{2})\) \(\approx\) \(1.002262567\)
\(L(\frac12)\) \(\approx\) \(1.002262567\)
\(L(3)\) 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 \( 1 + 7.18iT - 16T^{2} \)
5 \( 1 - 15.5iT - 625T^{2} \)
7 \( 1 + 43.6T + 2.40e3T^{2} \)
11 \( 1 - 1.21iT - 1.46e4T^{2} \)
13 \( 1 + 174.T + 2.85e4T^{2} \)
17 \( 1 - 372. iT - 8.35e4T^{2} \)
19 \( 1 - 70.8T + 1.30e5T^{2} \)
23 \( 1 - 65.1iT - 2.79e5T^{2} \)
29 \( 1 + 412. iT - 7.07e5T^{2} \)
31 \( 1 - 474.T + 9.23e5T^{2} \)
37 \( 1 - 403.T + 1.87e6T^{2} \)
41 \( 1 + 1.86e3iT - 2.82e6T^{2} \)
43 \( 1 + 261.T + 3.41e6T^{2} \)
47 \( 1 + 2.24e3iT - 4.87e6T^{2} \)
53 \( 1 - 879. iT - 7.89e6T^{2} \)
61 \( 1 - 718.T + 1.38e7T^{2} \)
67 \( 1 + 2.83e3T + 2.01e7T^{2} \)
71 \( 1 + 7.63e3iT - 2.54e7T^{2} \)
73 \( 1 + 6.75e3T + 2.83e7T^{2} \)
79 \( 1 - 1.26e3T + 3.89e7T^{2} \)
83 \( 1 - 2.47e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.24e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.31e4T + 8.85e7T^{2} \)
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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

−10.17892265281803734784298360761, −9.436783359221005303658793485029, −8.507738553580820110737230787325, −7.22360993045592067923881966032, −6.05270301145436316419051981158, −4.72375566408574600533957594475, −3.65715138663601974481486098639, −2.88395736727508189953624600366, −1.91731857586112647424066825893, −0.42659733520313916645582279271, 0.67060972558026328371262304028, 2.97634104697989788074186901074, 4.53567368748110734939109934761, 5.10671090906884959337795678574, 6.15994301219367202602719052424, 6.98816290656359683703702480675, 7.66986467730602069089298129515, 8.704279722932680441535092118962, 9.406758894298137249212660246289, 10.02469696402971312124178882655

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