LMFDB - L-function 2-33e2-3.2-c2-0-9

Dirichlet series

L(s) = 1

− 3.75i·2^-s − 10.0·4^-s + 7.83i·5^-s + 8.18·7^-s + 22.8i·8^-s + 29.4·10^-s + 2.87·13^-s − 30.6i·14^-s + 45.3·16^-s + 10.9i·17^-s − 26.5·19^-s − 79.0i·20^-s − 17.9i·23^-s − 36.4·25^-s − 10.7i·26^-s + ⋯

L(s) = 1

− 1.87i·2^-s − 2.52·4^-s + 1.56i·5^-s + 1.16·7^-s + 2.85i·8^-s + 2.94·10^-s + 0.221·13^-s − 2.19i·14^-s + 2.83·16^-s + 0.643i·17^-s − 1.39·19^-s − 3.95i·20^-s − 0.781i·23^-s − 1.45·25^-s − 0.414i·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

$L(\frac{3}{2})$	$\approx$	$0.6414625949$
$L(\frac12)$	$\approx$	$0.6414625949$
$L(2)$		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 + 3.75iT - 4T^{2} $
	5	$ 1 - 7.83iT - 25T^{2} $
	7	$ 1 - 8.18T + 49T^{2} $
	13	$ 1 - 2.87T + 169T^{2} $
	17	$ 1 - 10.9iT - 289T^{2} $
	19	$ 1 + 26.5T + 361T^{2} $
	23	$ 1 + 17.9iT - 529T^{2} $
	29	$ 1 - 37.5iT - 841T^{2} $
	31	$ 1 + 31.0T + 961T^{2} $
	37	$ 1 + 45.4T + 1.36e3T^{2} $
	41	$ 1 + 57.8iT - 1.68e3T^{2} $
	43	$ 1 - 9.29T + 1.84e3T^{2} $
	47	$ 1 - 24.4iT - 2.20e3T^{2} $
	53	$ 1 - 32.6iT - 2.80e3T^{2} $
	59	$ 1 - 8.48iT - 3.48e3T^{2} $
	61	$ 1 + 91.7T + 3.72e3T^{2} $
	67	$ 1 + 40.8T + 4.48e3T^{2} $
	71	$ 1 + 123. iT - 5.04e3T^{2} $
	73	$ 1 - 85.7T + 5.32e3T^{2} $
	79	$ 1 + 65.2T + 6.24e3T^{2} $
	83	$ 1 - 39.7iT - 6.88e3T^{2} $
	89	$ 1 - 27.6iT - 7.92e3T^{2} $
	97	$ 1 - 36.2T + 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

−10.48070331210425804764300542731, −9.134755953497749306019820801039, −8.505606406545281640518661009552, −7.51562403047894923875093541475, −6.35789897491072042241087152393, −5.12832985246856324825782406422, −4.12903283815530503492882794456, −3.36199643915309854267098979794, −2.32677650700877786892552541866, −1.62817435801681308075699775221, 0.19921488929492918187665252282, 1.56168166598395218433722889238, 4.04205335557734564007736833883, 4.69780904243611549264911691739, 5.29492811007661933152091367506, 6.04658372612420849667965056392, 7.16431719842269035363603956946, 8.020078764237145171376849919850, 8.440038106573029540516952520314, 9.073297309447745276448347384907

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.6414625949\)
\(L(\frac12)\)	\(\approx\)	\(0.6414625949\)
\(L(2)\)		not available
\(L(1)\)		not available

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