Properties

Label 2-531-1.1-c7-0-76
Degree $2$
Conductor $531$
Sign $1$
Analytic cond. $165.876$
Root an. cond. $12.8793$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13.6·2-s + 58.8·4-s + 207.·5-s + 882.·7-s + 945.·8-s − 2.83e3·10-s + 3.63e3·11-s + 208.·13-s − 1.20e4·14-s − 2.04e4·16-s + 1.31e4·17-s − 9.91e3·19-s + 1.21e4·20-s − 4.97e4·22-s + 3.77e4·23-s − 3.51e4·25-s − 2.84e3·26-s + 5.19e4·28-s + 9.80e4·29-s + 3.26e5·31-s + 1.58e5·32-s − 1.79e5·34-s + 1.83e5·35-s + 3.94e4·37-s + 1.35e5·38-s + 1.96e5·40-s + 7.19e5·41-s + ⋯
L(s)  = 1  − 1.20·2-s + 0.459·4-s + 0.742·5-s + 0.972·7-s + 0.652·8-s − 0.896·10-s + 0.823·11-s + 0.0262·13-s − 1.17·14-s − 1.24·16-s + 0.647·17-s − 0.331·19-s + 0.340·20-s − 0.995·22-s + 0.647·23-s − 0.449·25-s − 0.0317·26-s + 0.446·28-s + 0.746·29-s + 1.97·31-s + 0.855·32-s − 0.782·34-s + 0.721·35-s + 0.127·37-s + 0.400·38-s + 0.484·40-s + 1.63·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $1$
Analytic conductor: \(165.876\)
Root analytic conductor: \(12.8793\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.902221165\)
\(L(\frac12)\) \(\approx\) \(1.902221165\)
\(L(\frac{9}{2})\) 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 \( 1 - 2.05e5T \)
good2 \( 1 + 13.6T + 128T^{2} \)
5 \( 1 - 207.T + 7.81e4T^{2} \)
7 \( 1 - 882.T + 8.23e5T^{2} \)
11 \( 1 - 3.63e3T + 1.94e7T^{2} \)
13 \( 1 - 208.T + 6.27e7T^{2} \)
17 \( 1 - 1.31e4T + 4.10e8T^{2} \)
19 \( 1 + 9.91e3T + 8.93e8T^{2} \)
23 \( 1 - 3.77e4T + 3.40e9T^{2} \)
29 \( 1 - 9.80e4T + 1.72e10T^{2} \)
31 \( 1 - 3.26e5T + 2.75e10T^{2} \)
37 \( 1 - 3.94e4T + 9.49e10T^{2} \)
41 \( 1 - 7.19e5T + 1.94e11T^{2} \)
43 \( 1 + 8.78e5T + 2.71e11T^{2} \)
47 \( 1 - 1.58e5T + 5.06e11T^{2} \)
53 \( 1 - 1.83e6T + 1.17e12T^{2} \)
61 \( 1 - 3.49e6T + 3.14e12T^{2} \)
67 \( 1 + 1.00e6T + 6.06e12T^{2} \)
71 \( 1 + 3.31e5T + 9.09e12T^{2} \)
73 \( 1 + 5.27e5T + 1.10e13T^{2} \)
79 \( 1 + 1.50e6T + 1.92e13T^{2} \)
83 \( 1 + 2.48e6T + 2.71e13T^{2} \)
89 \( 1 + 6.37e6T + 4.42e13T^{2} \)
97 \( 1 + 4.33e6T + 8.07e13T^{2} \)
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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

−9.781320342904411474821560166019, −8.733976479783515106654766899739, −8.242216430595410023159572615717, −7.23398163978442357669756842356, −6.26480846159512422855896109271, −5.08992490189480703747660626873, −4.15149160562539407790997596600, −2.48781692902235961842788090481, −1.44541693781794139674085200592, −0.822318140945129499003997876317, 0.822318140945129499003997876317, 1.44541693781794139674085200592, 2.48781692902235961842788090481, 4.15149160562539407790997596600, 5.08992490189480703747660626873, 6.26480846159512422855896109271, 7.23398163978442357669756842356, 8.242216430595410023159572615717, 8.733976479783515106654766899739, 9.781320342904411474821560166019

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