LMFDB - L-function 2-33e2-3.2-c2-0-67

Dirichlet series

L(s) = 1

− 1.60i·2^-s + 1.41·4^-s − 6.21i·5^-s − 11.1·7^-s − 8.70i·8^-s − 9.98·10^-s + 17.7·13^-s + 17.9i·14^-s − 8.31·16^-s − 8.53i·17^-s − 16.4·19^-s − 8.80i·20^-s − 25.0i·23^-s − 13.5·25^-s − 28.5i·26^-s + ⋯

L(s) = 1

− 0.803i·2^-s + 0.354·4^-s − 1.24i·5^-s − 1.59·7^-s − 1.08i·8^-s − 0.998·10^-s + 1.36·13^-s + 1.27i·14^-s − 0.519·16^-s − 0.502i·17^-s − 0.866·19^-s − 0.440i·20^-s − 1.09i·23^-s − 0.543·25^-s − 1.09i·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1089$ = $3^{2} \cdot 11^{2}$
Sign:	$-0.577 - 0.816i$
Analytic conductor:	$29.6731$
Root analytic conductor:	$5.44730$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1089} (485, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1089,\ (\ :1),\ -0.577 - 0.816i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.9929842646$
$L(\frac12)$	$\approx$	$0.9929842646$
$L(2)$		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 $
	11	$ 1 $
good	2	$ 1 + 1.60iT - 4T^{2} $
	5	$ 1 + 6.21iT - 25T^{2} $
	7	$ 1 + 11.1T + 49T^{2} $
	13	$ 1 - 17.7T + 169T^{2} $
	17	$ 1 + 8.53iT - 289T^{2} $
	19	$ 1 + 16.4T + 361T^{2} $
	23	$ 1 + 25.0iT - 529T^{2} $
	29	$ 1 + 9.46iT - 841T^{2} $
	31	$ 1 + 46.1T + 961T^{2} $
	37	$ 1 + 37.4T + 1.36e3T^{2} $
	41	$ 1 - 51.8iT - 1.68e3T^{2} $
	43	$ 1 - 55.6T + 1.84e3T^{2} $
	47	$ 1 - 37.8iT - 2.20e3T^{2} $
	53	$ 1 - 58.5iT - 2.80e3T^{2} $
	59	$ 1 + 44.6iT - 3.48e3T^{2} $
	61	$ 1 - 10.1T + 3.72e3T^{2} $
	67	$ 1 + 58.3T + 4.48e3T^{2} $
	71	$ 1 - 75.1iT - 5.04e3T^{2} $
	73	$ 1 + 112.T + 5.32e3T^{2} $
	79	$ 1 + 52.2T + 6.24e3T^{2} $
	83	$ 1 + 18.0iT - 6.88e3T^{2} $
	89	$ 1 + 136. iT - 7.92e3T^{2} $
	97	$ 1 - 127.T + 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

−9.095474389182379903667363637585, −8.775059414464044771092325083843, −7.43213318872196013253899630659, −6.41262173273079685412555043457, −5.90730344318992853398573615057, −4.45627416852249953187167695337, −3.63157585435489495355779529089, −2.68996916129519062626356218950, −1.36342041027898595886336821621, −0.29113404827464644187027136212, 1.99841489883646292178510673962, 3.20552549403357173148217824936, 3.75702765973370736733531876199, 5.69111604798454262486085845730, 6.09236963332343341703504499045, 6.92735711899597938797907710216, 7.26855753483584990430123014627, 8.515081230865917396765457029079, 9.237895495899780944406041154302, 10.52949249252658712157281870883

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9929842646\)
\(L(\frac12)\)	\(\approx\)	\(0.9929842646\)
\(L(2)\)		not available
\(L(1)\)		not available

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