LMFDB - L-function 2-33e2-3.2-c2-0-54

Dirichlet series

L(s) = 1

− 2.98i·2^-s − 4.88·4^-s + 7.85i·5^-s − 2.80·7^-s + 2.63i·8^-s + 23.4·10^-s + 1.51·13^-s + 8.36i·14^-s − 11.6·16^-s + 26.3i·17^-s + 14.5·19^-s − 38.3i·20^-s − 25.0i·23^-s − 36.7·25^-s − 4.52i·26^-s + ⋯

L(s) = 1

− 1.49i·2^-s − 1.22·4^-s + 1.57i·5^-s − 0.401·7^-s + 0.328i·8^-s + 2.34·10^-s + 0.116·13^-s + 0.597i·14^-s − 0.730·16^-s + 1.55i·17^-s + 0.765·19^-s − 1.91i·20^-s − 1.09i·23^-s − 1.46·25^-s − 0.174i·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

$L(\frac{3}{2})$	$\approx$	$0.1360812737$
$L(\frac12)$	$\approx$	$0.1360812737$
$L(2)$		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 + 2.98iT - 4T^{2} $
	5	$ 1 - 7.85iT - 25T^{2} $
	7	$ 1 + 2.80T + 49T^{2} $
	13	$ 1 - 1.51T + 169T^{2} $
	17	$ 1 - 26.3iT - 289T^{2} $
	19	$ 1 - 14.5T + 361T^{2} $
	23	$ 1 + 25.0iT - 529T^{2} $
	29	$ 1 + 50.7iT - 841T^{2} $
	31	$ 1 + 51.6T + 961T^{2} $
	37	$ 1 + 19.5T + 1.36e3T^{2} $
	41	$ 1 + 30.9iT - 1.68e3T^{2} $
	43	$ 1 + 2.11T + 1.84e3T^{2} $
	47	$ 1 + 40.2iT - 2.20e3T^{2} $
	53	$ 1 + 41.1iT - 2.80e3T^{2} $
	59	$ 1 - 96.0iT - 3.48e3T^{2} $
	61	$ 1 - 2.89T + 3.72e3T^{2} $
	67	$ 1 + 88.2T + 4.48e3T^{2} $
	71	$ 1 - 11.3iT - 5.04e3T^{2} $
	73	$ 1 + 104.T + 5.32e3T^{2} $
	79	$ 1 + 16.7T + 6.24e3T^{2} $
	83	$ 1 - 22.7iT - 6.88e3T^{2} $
	89	$ 1 + 30.7iT - 7.92e3T^{2} $
	97	$ 1 + 28.5T + 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

−9.556534374075218051749732506269, −8.569551554061612109563249425426, −7.41327018510401194937083577335, −6.60488777527070622302166617161, −5.77456985363277428613645674863, −4.14560787135148242608288867999, −3.50412525744065463689413815777, −2.67152241587798102592253153184, −1.78036059341324792364733088959, −0.04113872671198303554959569150, 1.41280971491162689703113427857, 3.29644797980223587543413495172, 4.68883293124856017752460672951, 5.19860419520331321724964082483, 5.85709756439076968684337024954, 7.06048170975113229254622245983, 7.56290542704368721696116678561, 8.502771470814328775626705727131, 9.280239711076974213455646844231, 9.493913495081630883601548005553

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.1360812737\)
\(L(\frac12)\)	\(\approx\)	\(0.1360812737\)
\(L(2)\)		not available
\(L(1)\)		not available

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