LMFDB - L-function 2-33e2-33.32-c1-0-15

Dirichlet series

L(s) = 1

− 2.35·2^-s + 3.52·4^-s − 2.24i·5^-s + 4.05i·7^-s − 3.58·8^-s + 5.28i·10^-s − 4.65i·13^-s − 9.52i·14^-s + 1.38·16^-s + 0.0762·17^-s + 2.39i·19^-s − 7.92i·20^-s − 3.22i·23^-s − 0.0541·25^-s + 10.9i·26^-s + ⋯

L(s) = 1

− 1.66·2^-s + 1.76·4^-s − 1.00i·5^-s + 1.53i·7^-s − 1.26·8^-s + 1.67i·10^-s − 1.29i·13^-s − 2.54i·14^-s + 0.345·16^-s + 0.0185·17^-s + 0.549i·19^-s − 1.77i·20^-s − 0.672i·23^-s − 0.0108·25^-s + 2.14i·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 + 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1089$ = $3^{2} \cdot 11^{2}$
Sign:	$0.742 + 0.670i$
Analytic conductor:	$8.69570$
Root analytic conductor:	$2.94884$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1089} (1088, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1089,\ (\ :1/2),\ 0.742 + 0.670i)$

Particular Values

$L(1)$	$\approx$	$0.6549100374$
$L(\frac12)$	$\approx$	$0.6549100374$
$L(\frac{3}{2})$		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 + 2.35T + 2T^{2} $
	5	$ 1 + 2.24iT - 5T^{2} $
	7	$ 1 - 4.05iT - 7T^{2} $
	13	$ 1 + 4.65iT - 13T^{2} $
	17	$ 1 - 0.0762T + 17T^{2} $
	19	$ 1 - 2.39iT - 19T^{2} $
	23	$ 1 + 3.22iT - 23T^{2} $
	29	$ 1 - 1.83T + 29T^{2} $
	31	$ 1 - 1.67T + 31T^{2} $
	37	$ 1 - 7.26T + 37T^{2} $
	41	$ 1 + 8.44T + 41T^{2} $
	43	$ 1 + 4.28iT - 43T^{2} $
	47	$ 1 - 6.21iT - 47T^{2} $
	53	$ 1 - 1.22iT - 53T^{2} $
	59	$ 1 + 0.580iT - 59T^{2} $
	61	$ 1 + 3.78iT - 61T^{2} $
	67	$ 1 - 12.9T + 67T^{2} $
	71	$ 1 + 1.12iT - 71T^{2} $
	73	$ 1 + 13.3iT - 73T^{2} $
	79	$ 1 + 0.659iT - 79T^{2} $
	83	$ 1 - 10.2T + 83T^{2} $
	89	$ 1 - 6.58iT - 89T^{2} $
	97	$ 1 - 16.4T + 97T^{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

−9.552349670080952358068660478156, −8.888273043708382110795207780242, −8.296623360199941562867205296402, −7.85977757765401231228274141180, −6.52505248910018707448622996699, −5.67387098294094112822577897240, −4.79316296155433700274429002888, −2.98896708997841357777642368798, −1.95022436284174280585114856610, −0.66252041845139694383057129592, 0.968933882180737361786210759237, 2.22910118221352193450796195025, 3.51391495750073009017519728822, 4.62018570165626904119412241485, 6.42955551275307902723920047863, 6.93317418990346560846888670190, 7.44116879643350110344043658004, 8.322867299329349013755625697181, 9.280705063108255934688041643865, 10.01479153894407357939933273644

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6549100374\)
\(L(\frac12)\)	\(\approx\)	\(0.6549100374\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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