LMFDB - L-function 2-33e2-33.32-c1-0-6

Dirichlet series

L(s) = 1

− 0.688·2^-s − 1.52·4^-s + 0.0401i·5^-s − 0.246i·7^-s + 2.42·8^-s − 0.0276i·10^-s + 2.30i·13^-s + 0.169i·14^-s + 1.38·16^-s − 4.27·17^-s − 6.20i·19^-s − 0.0612i·20^-s + 6.79i·23^-s + 4.99·25^-s − 1.58i·26^-s + ⋯

L(s) = 1

− 0.486·2^-s − 0.763·4^-s + 0.0179i·5^-s − 0.0932i·7^-s + 0.858·8^-s − 0.00872i·10^-s + 0.638i·13^-s + 0.0454i·14^-s + 0.345·16^-s − 1.03·17^-s − 1.42i·19^-s − 0.0136i·20^-s + 1.41i·23^-s + 0.999·25^-s − 0.310i·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1089$ = $3^{2} \cdot 11^{2}$
Sign:	$0.384 - 0.923i$
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:	$(2,\ 1089,\ (\ :1/2),\ 0.384 - 0.923i)$

Particular Values

$L(1)$	$\approx$	$0.7941804466$
$L(\frac12)$	$\approx$	$0.7941804466$
$L(\frac{3}{2})$		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 + 0.688T + 2T^{2} $
	5	$ 1 - 0.0401iT - 5T^{2} $
	7	$ 1 + 0.246iT - 7T^{2} $
	13	$ 1 - 2.30iT - 13T^{2} $
	17	$ 1 + 4.27T + 17T^{2} $
	19	$ 1 + 6.20iT - 19T^{2} $
	23	$ 1 - 6.79iT - 23T^{2} $
	29	$ 1 + 5.59T + 29T^{2} $
	31	$ 1 - 4.79T + 31T^{2} $
	37	$ 1 + 4.03T + 37T^{2} $
	41	$ 1 - 9.60T + 41T^{2} $
	43	$ 1 - 1.03iT - 43T^{2} $
	47	$ 1 - 11.1iT - 47T^{2} $
	53	$ 1 - 8.96iT - 53T^{2} $
	59	$ 1 - 2.78iT - 59T^{2} $
	61	$ 1 - 8.48iT - 61T^{2} $
	67	$ 1 - 7.94T + 67T^{2} $
	71	$ 1 - 3.32iT - 71T^{2} $
	73	$ 1 - 11.8iT - 73T^{2} $
	79	$ 1 - 3.01iT - 79T^{2} $
	83	$ 1 + 5.29T + 83T^{2} $
	89	$ 1 - 8.54iT - 89T^{2} $
	97	$ 1 + 3.02T + 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

−9.749590656697379537345782249528, −9.151847031324253111812193096987, −8.661944624461309309707127280918, −7.53239586995472275178768725296, −6.93045106611899315665111024350, −5.71183822392350844525876943205, −4.69493841996800004765734577326, −4.03858853264800273785022775858, −2.61713481083898514510419594548, −1.11819915132474521782161659814, 0.51408885130907543948359990438, 2.07359307152150981892691724188, 3.53231076987696099132894774539, 4.48285572926991813837212151585, 5.35012247395157256996385435979, 6.37408112398852213898326472885, 7.40735584071940610329828522591, 8.341813322465757819307096544023, 8.740319946165314148553589741992, 9.709259270341238189946421873510

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7941804466\)
\(L(\frac12)\)	\(\approx\)	\(0.7941804466\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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