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Properties

Label 2-531-59.58-c2-0-20
Degree $2$
Conductor $531$
Sign $0.904 - 0.425i$
Analytic cond. $14.4687$
Root an. cond. $3.80377$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.537i·2-s + 3.71·4-s + 0.803·5-s + 5.11·7-s − 4.14i·8-s − 0.431i·10-s + 17.7i·11-s + 24.6i·13-s − 2.74i·14-s + 12.6·16-s − 18.2·17-s + 28.5·19-s + 2.98·20-s + 9.54·22-s − 11.8i·23-s + ⋯
L(s)  = 1  − 0.268i·2-s + 0.927·4-s + 0.160·5-s + 0.730·7-s − 0.518i·8-s − 0.0431i·10-s + 1.61i·11-s + 1.89i·13-s − 0.196i·14-s + 0.788·16-s − 1.07·17-s + 1.50·19-s + 0.149·20-s + 0.433·22-s − 0.515i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.904 - 0.425i$
Analytic conductor: \(14.4687\)
Root analytic conductor: \(3.80377\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :1),\ 0.904 - 0.425i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.438204587\)
\(L(\frac12)\) \(\approx\) \(2.438204587\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 + (-53.3 + 25.1i)T \)
good2 \( 1 + 0.537iT - 4T^{2} \)
5 \( 1 - 0.803T + 25T^{2} \)
7 \( 1 - 5.11T + 49T^{2} \)
11 \( 1 - 17.7iT - 121T^{2} \)
13 \( 1 - 24.6iT - 169T^{2} \)
17 \( 1 + 18.2T + 289T^{2} \)
19 \( 1 - 28.5T + 361T^{2} \)
23 \( 1 + 11.8iT - 529T^{2} \)
29 \( 1 + 9.01T + 841T^{2} \)
31 \( 1 - 4.94iT - 961T^{2} \)
37 \( 1 + 39.7iT - 1.36e3T^{2} \)
41 \( 1 - 38.0T + 1.68e3T^{2} \)
43 \( 1 - 19.2iT - 1.84e3T^{2} \)
47 \( 1 - 65.6iT - 2.20e3T^{2} \)
53 \( 1 - 40.1T + 2.80e3T^{2} \)
61 \( 1 + 110. iT - 3.72e3T^{2} \)
67 \( 1 - 30.7iT - 4.48e3T^{2} \)
71 \( 1 - 95.0T + 5.04e3T^{2} \)
73 \( 1 + 71.2iT - 5.32e3T^{2} \)
79 \( 1 + 13.3T + 6.24e3T^{2} \)
83 \( 1 + 142. iT - 6.88e3T^{2} \)
89 \( 1 + 128. iT - 7.92e3T^{2} \)
97 \( 1 - 97.1iT - 9.40e3T^{2} \)
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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

−10.92133810619941520071163248892, −9.736685867623044593947869560200, −9.235187418608121752900942774036, −7.74661540954636506719072539930, −7.11073453254088777583295217756, −6.29614445802485597666455345174, −4.89688064156859300070750394886, −4.01534697606554776334575969620, −2.28592371281616232342462279810, −1.67627561284129541613154920242, 0.998321709665383259571747120088, 2.57307947872753627832815339280, 3.55954927081647810247519464092, 5.45977847595469692843635578377, 5.68561789328781720754368238560, 6.99815422177905312382758053080, 7.967042203222661921464078689032, 8.429330973331857617861413758255, 9.815328413737606772029334526403, 10.79829000449962129367074445912

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