Properties

Label 2-531-1.1-c3-0-61
Degree $2$
Conductor $531$
Sign $-1$
Analytic cond. $31.3300$
Root an. cond. $5.59732$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.06·2-s + 1.36·4-s − 0.675·5-s − 3.38·7-s − 20.3·8-s − 2.06·10-s + 8.79·11-s + 59.4·13-s − 10.3·14-s − 73.0·16-s − 31.2·17-s − 77.5·19-s − 0.921·20-s + 26.9·22-s − 73.1·23-s − 124.·25-s + 181.·26-s − 4.62·28-s + 61.5·29-s − 278.·31-s − 61.1·32-s − 95.5·34-s + 2.28·35-s − 326.·37-s − 237.·38-s + 13.7·40-s − 453.·41-s + ⋯
L(s)  = 1  + 1.08·2-s + 0.170·4-s − 0.0603·5-s − 0.182·7-s − 0.897·8-s − 0.0653·10-s + 0.240·11-s + 1.26·13-s − 0.197·14-s − 1.14·16-s − 0.445·17-s − 0.936·19-s − 0.0102·20-s + 0.260·22-s − 0.662·23-s − 0.996·25-s + 1.37·26-s − 0.0311·28-s + 0.394·29-s − 1.61·31-s − 0.337·32-s − 0.482·34-s + 0.0110·35-s − 1.45·37-s − 1.01·38-s + 0.0541·40-s − 1.72·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-1$
Analytic conductor: \(31.3300\)
Root analytic conductor: \(5.59732\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 531,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 59T \)
good2 \( 1 - 3.06T + 8T^{2} \)
5 \( 1 + 0.675T + 125T^{2} \)
7 \( 1 + 3.38T + 343T^{2} \)
11 \( 1 - 8.79T + 1.33e3T^{2} \)
13 \( 1 - 59.4T + 2.19e3T^{2} \)
17 \( 1 + 31.2T + 4.91e3T^{2} \)
19 \( 1 + 77.5T + 6.85e3T^{2} \)
23 \( 1 + 73.1T + 1.21e4T^{2} \)
29 \( 1 - 61.5T + 2.43e4T^{2} \)
31 \( 1 + 278.T + 2.97e4T^{2} \)
37 \( 1 + 326.T + 5.06e4T^{2} \)
41 \( 1 + 453.T + 6.89e4T^{2} \)
43 \( 1 + 174.T + 7.95e4T^{2} \)
47 \( 1 - 480.T + 1.03e5T^{2} \)
53 \( 1 - 185.T + 1.48e5T^{2} \)
61 \( 1 - 807.T + 2.26e5T^{2} \)
67 \( 1 + 129.T + 3.00e5T^{2} \)
71 \( 1 - 272.T + 3.57e5T^{2} \)
73 \( 1 + 610.T + 3.89e5T^{2} \)
79 \( 1 - 886.T + 4.93e5T^{2} \)
83 \( 1 + 51.4T + 5.71e5T^{2} \)
89 \( 1 + 182.T + 7.04e5T^{2} \)
97 \( 1 + 1.35e3T + 9.12e5T^{2} \)
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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

−10.10002726510585210912245591201, −8.949081116962735840669655549841, −8.376533826937343132435225668926, −6.89258736654106819569797434542, −6.11454650173919116577272625845, −5.25875946944311371858879855021, −4.04751748489501243914096474368, −3.50290173008754511860339986995, −1.95173922906088072905675728856, 0, 1.95173922906088072905675728856, 3.50290173008754511860339986995, 4.04751748489501243914096474368, 5.25875946944311371858879855021, 6.11454650173919116577272625845, 6.89258736654106819569797434542, 8.376533826937343132435225668926, 8.949081116962735840669655549841, 10.10002726510585210912245591201

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