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Properties

Label 2-531-3.2-c4-0-24
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  − 6.47i·2-s − 25.8·4-s + 30.8i·5-s + 46.2·7-s + 63.8i·8-s + 199.·10-s − 197. i·11-s − 90.6·13-s − 299. i·14-s − 0.653·16-s + 534. i·17-s + 313.·19-s − 799. i·20-s − 1.27e3·22-s + 67.2i·23-s + ⋯
L(s)  = 1  − 1.61i·2-s − 1.61·4-s + 1.23i·5-s + 0.944·7-s + 0.997i·8-s + 1.99·10-s − 1.62i·11-s − 0.536·13-s − 1.52i·14-s − 0.00255·16-s + 1.84i·17-s + 0.867·19-s − 1.99i·20-s − 2.63·22-s + 0.127i·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

\(L(\frac{5}{2})\) \(\approx\) \(1.792865288\)
\(L(\frac12)\) \(\approx\) \(1.792865288\)
\(L(3)\) 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 \( 1 + 6.47iT - 16T^{2} \)
5 \( 1 - 30.8iT - 625T^{2} \)
7 \( 1 - 46.2T + 2.40e3T^{2} \)
11 \( 1 + 197. iT - 1.46e4T^{2} \)
13 \( 1 + 90.6T + 2.85e4T^{2} \)
17 \( 1 - 534. iT - 8.35e4T^{2} \)
19 \( 1 - 313.T + 1.30e5T^{2} \)
23 \( 1 - 67.2iT - 2.79e5T^{2} \)
29 \( 1 - 398. iT - 7.07e5T^{2} \)
31 \( 1 + 1.16e3T + 9.23e5T^{2} \)
37 \( 1 + 154.T + 1.87e6T^{2} \)
41 \( 1 - 1.05e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.93e3T + 3.41e6T^{2} \)
47 \( 1 - 1.57e3iT - 4.87e6T^{2} \)
53 \( 1 - 527. iT - 7.89e6T^{2} \)
61 \( 1 - 2.38e3T + 1.38e7T^{2} \)
67 \( 1 - 6.67e3T + 2.01e7T^{2} \)
71 \( 1 + 3.24e3iT - 2.54e7T^{2} \)
73 \( 1 - 8.30e3T + 2.83e7T^{2} \)
79 \( 1 + 1.75e3T + 3.89e7T^{2} \)
83 \( 1 - 3.42e3iT - 4.74e7T^{2} \)
89 \( 1 - 2.36e3iT - 6.27e7T^{2} \)
97 \( 1 - 4.15e3T + 8.85e7T^{2} \)
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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

−10.72704910184389971637835863848, −9.596305711114403286860519069793, −8.581311272916770348085483302295, −7.69489145618558057883222358252, −6.38897635763893342619193943124, −5.27050680271970673323135007232, −3.85310175332092512612689046020, −3.20930642469841394230391760599, −2.15323668156497775327047086666, −1.04501028270632321271444596206, 0.54133531801326304894463288796, 2.07247559015204194726160691117, 4.34193997971279708194868672393, 5.03720542270421502407291581233, 5.38350684904611699120787358813, 7.05920213732899318100559663108, 7.43179979331889236946751164609, 8.316360972894663899489322967917, 9.264725318440830134652369642904, 9.716868519839392613992738741402

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