Properties

Label 2-531-59.58-c2-0-40
Degree $2$
Conductor $531$
Sign $-0.738 + 0.674i$
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  − 1.08i·2-s + 2.82·4-s − 3.33·5-s − 1.22·7-s − 7.40i·8-s + 3.62i·10-s + 1.21i·11-s − 12.6i·13-s + 1.33i·14-s + 3.23·16-s + 0.815·17-s − 7.74·19-s − 9.40·20-s + 1.32·22-s − 33.8i·23-s + ⋯
L(s)  = 1  − 0.543i·2-s + 0.705·4-s − 0.667·5-s − 0.175·7-s − 0.926i·8-s + 0.362i·10-s + 0.110i·11-s − 0.973i·13-s + 0.0951i·14-s + 0.202·16-s + 0.0479·17-s − 0.407·19-s − 0.470·20-s + 0.0601·22-s − 1.46i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.738 + 0.674i$
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: \((2,\ 531,\ (\ :1),\ -0.738 + 0.674i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.379308560\)
\(L(\frac12)\) \(\approx\) \(1.379308560\)
\(L(2)\) 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 \( 1 + (43.5 - 39.7i)T \)
good2 \( 1 + 1.08iT - 4T^{2} \)
5 \( 1 + 3.33T + 25T^{2} \)
7 \( 1 + 1.22T + 49T^{2} \)
11 \( 1 - 1.21iT - 121T^{2} \)
13 \( 1 + 12.6iT - 169T^{2} \)
17 \( 1 - 0.815T + 289T^{2} \)
19 \( 1 + 7.74T + 361T^{2} \)
23 \( 1 + 33.8iT - 529T^{2} \)
29 \( 1 + 7.53T + 841T^{2} \)
31 \( 1 + 7.71iT - 961T^{2} \)
37 \( 1 + 16.7iT - 1.36e3T^{2} \)
41 \( 1 - 19.0T + 1.68e3T^{2} \)
43 \( 1 + 48.9iT - 1.84e3T^{2} \)
47 \( 1 + 4.79iT - 2.20e3T^{2} \)
53 \( 1 - 48.7T + 2.80e3T^{2} \)
61 \( 1 + 87.6iT - 3.72e3T^{2} \)
67 \( 1 - 42.6iT - 4.48e3T^{2} \)
71 \( 1 - 20.2T + 5.04e3T^{2} \)
73 \( 1 + 45.6iT - 5.32e3T^{2} \)
79 \( 1 + 62.5T + 6.24e3T^{2} \)
83 \( 1 - 22.8iT - 6.88e3T^{2} \)
89 \( 1 - 55.8iT - 7.92e3T^{2} \)
97 \( 1 - 123. iT - 9.40e3T^{2} \)
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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

−10.53681590930392973138298116615, −9.661291785680430489685028781207, −8.423907373120456259035219841003, −7.60662259591129370900989032043, −6.70751976352612715844364490033, −5.71048644968505533209479366368, −4.27239038957388554752056826025, −3.28554605781136475405007623299, −2.19440289673321067225672158185, −0.51194596940299107784568935125, 1.72851770721700972225655318418, 3.16048690758936775753374396241, 4.33813224998593591868018194337, 5.61139272784680390965027191176, 6.51322759683024515773583889126, 7.36310377508322621466231233526, 8.043886396422117366818615627999, 9.071512935273913897608041568376, 10.08753197920124802458057895977, 11.33155026915151019452702783371

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