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Properties

Label 2-531-59.58-c2-0-25
Degree $2$
Conductor $531$
Sign $0.0793 - 0.996i$
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  + 1.61i·2-s + 1.39·4-s + 6.36·5-s + 6.66·7-s + 8.70i·8-s + 10.2i·10-s + 16.0i·11-s − 7.84i·13-s + 10.7i·14-s − 8.46·16-s − 18.9·17-s + 7.10·19-s + 8.88·20-s − 25.8·22-s − 33.6i·23-s + ⋯
L(s)  = 1  + 0.806i·2-s + 0.349·4-s + 1.27·5-s + 0.952·7-s + 1.08i·8-s + 1.02i·10-s + 1.45i·11-s − 0.603i·13-s + 0.768i·14-s − 0.529·16-s − 1.11·17-s + 0.374·19-s + 0.444·20-s − 1.17·22-s − 1.46i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0793 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0793 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.0793 - 0.996i$
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: \((2,\ 531,\ (\ :1),\ 0.0793 - 0.996i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.819335977\)
\(L(\frac12)\) \(\approx\) \(2.819335977\)
\(L(2)\) 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 \( 1 + (-4.68 + 58.8i)T \)
good2 \( 1 - 1.61iT - 4T^{2} \)
5 \( 1 - 6.36T + 25T^{2} \)
7 \( 1 - 6.66T + 49T^{2} \)
11 \( 1 - 16.0iT - 121T^{2} \)
13 \( 1 + 7.84iT - 169T^{2} \)
17 \( 1 + 18.9T + 289T^{2} \)
19 \( 1 - 7.10T + 361T^{2} \)
23 \( 1 + 33.6iT - 529T^{2} \)
29 \( 1 - 46.2T + 841T^{2} \)
31 \( 1 - 29.5iT - 961T^{2} \)
37 \( 1 - 1.91iT - 1.36e3T^{2} \)
41 \( 1 + 46.4T + 1.68e3T^{2} \)
43 \( 1 + 21.6iT - 1.84e3T^{2} \)
47 \( 1 + 75.3iT - 2.20e3T^{2} \)
53 \( 1 + 19.5T + 2.80e3T^{2} \)
61 \( 1 - 41.0iT - 3.72e3T^{2} \)
67 \( 1 - 90.1iT - 4.48e3T^{2} \)
71 \( 1 + 57.1T + 5.04e3T^{2} \)
73 \( 1 + 69.6iT - 5.32e3T^{2} \)
79 \( 1 - 118.T + 6.24e3T^{2} \)
83 \( 1 - 86.3iT - 6.88e3T^{2} \)
89 \( 1 - 38.4iT - 7.92e3T^{2} \)
97 \( 1 - 24.0iT - 9.40e3T^{2} \)
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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

−10.56958205747035646512133366451, −10.12449815735035456421605255714, −8.837450945688934365663686358046, −8.134611375708649685947527219812, −6.96613168695360618124004873206, −6.47432886667164935656914823630, −5.24481134607483225883336471337, −4.72163135388131994341377633013, −2.51939619096880640821651590491, −1.74390007606598980137827724786, 1.22412211194986054916924810845, 2.13332081004550938220364777425, 3.24581263225529169185936024196, 4.66044547320948366667026207347, 5.87903895497571675969306340075, 6.53491420625986995295053367957, 7.81888111323723724557895274830, 8.930359801162261200793346225487, 9.669933265421590582469737984025, 10.60150999938503964810923690074

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