Properties

Label 2-531-59.58-c4-0-90
Degree $2$
Conductor $531$
Sign $-0.989 + 0.142i$
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  − 4.05i·2-s − 0.408·4-s + 16.4·5-s + 47.3·7-s − 63.1i·8-s − 66.4i·10-s − 25.6i·11-s + 105. i·13-s − 191. i·14-s − 262.·16-s − 441.·17-s − 560.·19-s − 6.69·20-s − 103.·22-s − 764. i·23-s + ⋯
L(s)  = 1  − 1.01i·2-s − 0.0255·4-s + 0.656·5-s + 0.965·7-s − 0.986i·8-s − 0.664i·10-s − 0.211i·11-s + 0.624i·13-s − 0.977i·14-s − 1.02·16-s − 1.52·17-s − 1.55·19-s − 0.0167·20-s − 0.214·22-s − 1.44i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.989 + 0.142i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.989 + 0.142i$
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: \((2,\ 531,\ (\ :2),\ -0.989 + 0.142i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.126229147\)
\(L(\frac12)\) \(\approx\) \(2.126229147\)
\(L(3)\) 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 \( 1 + (-3.44e3 + 496. i)T \)
good2 \( 1 + 4.05iT - 16T^{2} \)
5 \( 1 - 16.4T + 625T^{2} \)
7 \( 1 - 47.3T + 2.40e3T^{2} \)
11 \( 1 + 25.6iT - 1.46e4T^{2} \)
13 \( 1 - 105. iT - 2.85e4T^{2} \)
17 \( 1 + 441.T + 8.35e4T^{2} \)
19 \( 1 + 560.T + 1.30e5T^{2} \)
23 \( 1 + 764. iT - 2.79e5T^{2} \)
29 \( 1 - 1.38e3T + 7.07e5T^{2} \)
31 \( 1 + 950. iT - 9.23e5T^{2} \)
37 \( 1 - 632. iT - 1.87e6T^{2} \)
41 \( 1 + 1.02e3T + 2.82e6T^{2} \)
43 \( 1 + 2.95e3iT - 3.41e6T^{2} \)
47 \( 1 + 4.32e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.11e3T + 7.89e6T^{2} \)
61 \( 1 - 4.27e3iT - 1.38e7T^{2} \)
67 \( 1 + 867. iT - 2.01e7T^{2} \)
71 \( 1 - 236.T + 2.54e7T^{2} \)
73 \( 1 + 6.21e3iT - 2.83e7T^{2} \)
79 \( 1 - 8.91e3T + 3.89e7T^{2} \)
83 \( 1 + 1.08e4iT - 4.74e7T^{2} \)
89 \( 1 + 6.26e3iT - 6.27e7T^{2} \)
97 \( 1 - 7.89e3iT - 8.85e7T^{2} \)
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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

−10.27987717790780156147761383533, −8.956095987910266032216993160803, −8.394234567017378421602425518980, −6.85112717665453787120586799072, −6.28396844639780520625878052327, −4.75772155443203869313026775457, −4.00234741100444117960276185874, −2.27142805049848203300396172725, −2.03057059674028894106696935191, −0.47853461608873808072279102465, 1.60441828950474929079218403447, 2.55568348196194814855375517988, 4.42311102748954414450688957069, 5.24460451316042251382848854470, 6.21302912317972973758881396428, 6.89887799806418384410250587949, 8.036648602296318123069952415617, 8.512802809627875085079447150785, 9.635233854625903742943375867933, 10.81610717002480686868197299517

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