LMFDB - L-function 2-33e2-3.2-c2-0-1

Dirichlet series

L(s) = 1

− 2.99i·2^-s − 4.96·4^-s + 1.61i·5^-s − 3.32·7^-s + 2.90i·8^-s + 4.82·10^-s + 7.56·13^-s + 9.97i·14^-s − 11.1·16^-s − 28.3i·17^-s − 26.0·19^-s − 8.00i·20^-s + 19.5i·23^-s + 22.4·25^-s − 22.6i·26^-s + ⋯

L(s) = 1

− 1.49i·2^-s − 1.24·4^-s + 0.322i·5^-s − 0.475·7^-s + 0.362i·8^-s + 0.482·10^-s + 0.581·13^-s + 0.712i·14^-s − 0.698·16^-s − 1.66i·17^-s − 1.37·19^-s − 0.400i·20^-s + 0.850i·23^-s + 0.896·25^-s − 0.871i·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

$L(\frac{3}{2})$	$\approx$	$0.1300813558$
$L(\frac12)$	$\approx$	$0.1300813558$
$L(2)$		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 + 2.99iT - 4T^{2} $
	5	$ 1 - 1.61iT - 25T^{2} $
	7	$ 1 + 3.32T + 49T^{2} $
	13	$ 1 - 7.56T + 169T^{2} $
	17	$ 1 + 28.3iT - 289T^{2} $
	19	$ 1 + 26.0T + 361T^{2} $
	23	$ 1 - 19.5iT - 529T^{2} $
	29	$ 1 - 2.15iT - 841T^{2} $
	31	$ 1 + 9.22T + 961T^{2} $
	37	$ 1 + 67.5T + 1.36e3T^{2} $
	41	$ 1 - 29.0iT - 1.68e3T^{2} $
	43	$ 1 - 0.719T + 1.84e3T^{2} $
	47	$ 1 - 24.5iT - 2.20e3T^{2} $
	53	$ 1 - 5.79iT - 2.80e3T^{2} $
	59	$ 1 - 73.9iT - 3.48e3T^{2} $
	61	$ 1 - 72.1T + 3.72e3T^{2} $
	67	$ 1 + 79.8T + 4.48e3T^{2} $
	71	$ 1 - 107. iT - 5.04e3T^{2} $
	73	$ 1 + 90.3T + 5.32e3T^{2} $
	79	$ 1 - 114.T + 6.24e3T^{2} $
	83	$ 1 - 125. iT - 6.88e3T^{2} $
	89	$ 1 + 83.3iT - 7.92e3T^{2} $
	97	$ 1 + 78.0T + 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

−9.991971839407682583799620825553, −9.233089155357903755053536606624, −8.569230634568019519458201477374, −7.20441566629093808690673997840, −6.53257753829777911327376265159, −5.25021326262632759999494527048, −4.20828631474685970801105989261, −3.26913201978142108102621285522, −2.55813658672398936705399678169, −1.31803972594885055896761285253, 0.03996517539602733269553786772, 1.93392039720923314851924845695, 3.60675870354707340129832086711, 4.55296651669637571134679941327, 5.51295871063408895844078455077, 6.41501443437068909162527650157, 6.72911448298558300620854565757, 7.953317202931829876805557146665, 8.598373663320143515782313282050, 9.011625174692596310523846086741

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.1300813558\)
\(L(\frac12)\)	\(\approx\)	\(0.1300813558\)
\(L(2)\)		not available
\(L(1)\)		not available

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