Properties

Label 2-531-1.1-c5-0-31
Degree $2$
Conductor $531$
Sign $1$
Analytic cond. $85.1638$
Root an. cond. $9.22842$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.20·2-s − 21.7·4-s − 26.7·5-s + 39.0·7-s + 172.·8-s + 85.6·10-s + 606.·11-s − 161.·13-s − 125.·14-s + 142.·16-s + 1.65e3·17-s + 882.·19-s + 580.·20-s − 1.94e3·22-s − 2.92e3·23-s − 2.41e3·25-s + 518.·26-s − 847.·28-s + 7.06e3·29-s − 2.25e3·31-s − 5.96e3·32-s − 5.29e3·34-s − 1.04e3·35-s − 7.56e3·37-s − 2.82e3·38-s − 4.60e3·40-s + 1.67e4·41-s + ⋯
L(s)  = 1  − 0.566·2-s − 0.678·4-s − 0.478·5-s + 0.301·7-s + 0.951·8-s + 0.270·10-s + 1.51·11-s − 0.265·13-s − 0.170·14-s + 0.139·16-s + 1.38·17-s + 0.560·19-s + 0.324·20-s − 0.856·22-s − 1.15·23-s − 0.771·25-s + 0.150·26-s − 0.204·28-s + 1.55·29-s − 0.421·31-s − 1.03·32-s − 0.785·34-s − 0.143·35-s − 0.908·37-s − 0.317·38-s − 0.454·40-s + 1.55·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $1$
Analytic conductor: \(85.1638\)
Root analytic conductor: \(9.22842\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.321750838\)
\(L(\frac12)\) \(\approx\) \(1.321750838\)
\(L(\frac{7}{2})\) 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 \( 1 + 3.48e3T \)
good2 \( 1 + 3.20T + 32T^{2} \)
5 \( 1 + 26.7T + 3.12e3T^{2} \)
7 \( 1 - 39.0T + 1.68e4T^{2} \)
11 \( 1 - 606.T + 1.61e5T^{2} \)
13 \( 1 + 161.T + 3.71e5T^{2} \)
17 \( 1 - 1.65e3T + 1.41e6T^{2} \)
19 \( 1 - 882.T + 2.47e6T^{2} \)
23 \( 1 + 2.92e3T + 6.43e6T^{2} \)
29 \( 1 - 7.06e3T + 2.05e7T^{2} \)
31 \( 1 + 2.25e3T + 2.86e7T^{2} \)
37 \( 1 + 7.56e3T + 6.93e7T^{2} \)
41 \( 1 - 1.67e4T + 1.15e8T^{2} \)
43 \( 1 + 4.50e3T + 1.47e8T^{2} \)
47 \( 1 + 8.40e3T + 2.29e8T^{2} \)
53 \( 1 + 1.10e4T + 4.18e8T^{2} \)
61 \( 1 - 3.47e4T + 8.44e8T^{2} \)
67 \( 1 + 5.72e4T + 1.35e9T^{2} \)
71 \( 1 + 26.9T + 1.80e9T^{2} \)
73 \( 1 - 5.44e4T + 2.07e9T^{2} \)
79 \( 1 - 9.31e4T + 3.07e9T^{2} \)
83 \( 1 - 8.96e4T + 3.93e9T^{2} \)
89 \( 1 - 1.28e5T + 5.58e9T^{2} \)
97 \( 1 + 2.66e4T + 8.58e9T^{2} \)
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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

−9.793110422451868387782481867605, −9.303337803539264084120571349559, −8.176680914829408779756399545401, −7.73267376802423352967762319058, −6.52506849131434740032877624638, −5.31654307206667007447195059025, −4.25543155996953000858868847705, −3.47765007671054179789174802162, −1.62032381193267384516347852672, −0.67720617584632738513389113093, 0.67720617584632738513389113093, 1.62032381193267384516347852672, 3.47765007671054179789174802162, 4.25543155996953000858868847705, 5.31654307206667007447195059025, 6.52506849131434740032877624638, 7.73267376802423352967762319058, 8.176680914829408779756399545401, 9.303337803539264084120571349559, 9.793110422451868387782481867605

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