LMFDB - L-function 2-531-59.58-c2-0-9

Dirichlet series

L(s) = 1

− 3.65i·2^-s − 9.37·4^-s − 0.395·5^-s − 6.20·7^-s + 19.6i·8^-s + 1.44i·10^-s + 8.54i·11^-s − 0.0887i·13^-s + 22.7i·14^-s + 34.4·16^-s + 8.10·17^-s + 10.5·19^-s + 3.71·20^-s + 31.2·22^-s + 14.0i·23^-s + ⋯

L(s) = 1

− 1.82i·2^-s − 2.34·4^-s − 0.0791·5^-s − 0.886·7^-s + 2.45i·8^-s + 0.144i·10^-s + 0.777i·11^-s − 0.00682i·13^-s + 1.62i·14^-s + 2.15·16^-s + 0.477·17^-s + 0.552·19^-s + 0.185·20^-s + 1.42·22^-s + 0.610i·23^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$531$ = $3^{2} \cdot 59$
Sign:	$0.703 + 0.710i$
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:	$(2,\ 531,\ (\ :1),\ 0.703 + 0.710i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9452246682$
$L(\frac12)$	$\approx$	$0.9452246682$
$L(2)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	59	$ 1 + (-41.5 - 41.9i)T $
good	2	$ 1 + 3.65iT - 4T^{2} $
	5	$ 1 + 0.395T + 25T^{2} $
	7	$ 1 + 6.20T + 49T^{2} $
	11	$ 1 - 8.54iT - 121T^{2} $
	13	$ 1 + 0.0887iT - 169T^{2} $
	17	$ 1 - 8.10T + 289T^{2} $
	19	$ 1 - 10.5T + 361T^{2} $
	23	$ 1 - 14.0iT - 529T^{2} $
	29	$ 1 - 56.9T + 841T^{2} $
	31	$ 1 - 0.471iT - 961T^{2} $
	37	$ 1 - 43.8iT - 1.36e3T^{2} $
	41	$ 1 + 13.8T + 1.68e3T^{2} $
	43	$ 1 + 53.7iT - 1.84e3T^{2} $
	47	$ 1 - 34.1iT - 2.20e3T^{2} $
	53	$ 1 - 79.3T + 2.80e3T^{2} $
	61	$ 1 + 45.7iT - 3.72e3T^{2} $
	67	$ 1 - 104. iT - 4.48e3T^{2} $
	71	$ 1 + 74.0T + 5.04e3T^{2} $
	73	$ 1 - 109. iT - 5.32e3T^{2} $
	79	$ 1 + 74.2T + 6.24e3T^{2} $
	83	$ 1 + 88.8iT - 6.88e3T^{2} $
	89	$ 1 - 142. iT - 7.92e3T^{2} $
	97	$ 1 - 81.4iT - 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

−10.21972315799251635163268856982, −10.09027872184737097312496058061, −9.203860295364281043935689227500, −8.214257173541693678808030966898, −6.93397092154878638962511483230, −5.51092384801931808331336819437, −4.38812498109501155425157328791, −3.43251654357250923748135335735, −2.50839683200871557936301678473, −1.12691216548067634770743792415, 0.45018850912162817115441266898, 3.21000358915362758619744551785, 4.37481816473240439043860962064, 5.53475848276359375383227087514, 6.22628666303847627115148768079, 7.00136374237757248529337350642, 7.957343313717047866486344691126, 8.669145154411268138703289980187, 9.559059058268058706345154033909, 10.34051985283633923294051325409

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9452246682\)
\(L(\frac12)\)	\(\approx\)	\(0.9452246682\)
\(L(2)\)		not available
\(L(1)\)		not available

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