LMFDB - L-function 2-33e2-3.2-c2-0-26

Dirichlet series

L(s) = 1

− 3.44i·2^-s − 7.87·4^-s + 6.27i·5^-s − 8.87·7^-s + 13.3i·8^-s + 21.6·10^-s − 24.7·13^-s + 30.5i·14^-s + 14.4·16^-s − 4.68i·17^-s + 29.2·19^-s − 49.3i·20^-s + 27.7i·23^-s − 14.3·25^-s + 85.2i·26^-s + ⋯

L(s) = 1

− 1.72i·2^-s − 1.96·4^-s + 1.25i·5^-s − 1.26·7^-s + 1.66i·8^-s + 2.16·10^-s − 1.90·13^-s + 2.18i·14^-s + 0.905·16^-s − 0.275i·17^-s + 1.53·19^-s − 2.46i·20^-s + 1.20i·23^-s − 0.574·25^-s + 3.27i·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

$L(\frac{3}{2})$	$\approx$	$0.9451036990$
$L(\frac12)$	$\approx$	$0.9451036990$
$L(2)$		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 + 3.44iT - 4T^{2} $
	5	$ 1 - 6.27iT - 25T^{2} $
	7	$ 1 + 8.87T + 49T^{2} $
	13	$ 1 + 24.7T + 169T^{2} $
	17	$ 1 + 4.68iT - 289T^{2} $
	19	$ 1 - 29.2T + 361T^{2} $
	23	$ 1 - 27.7iT - 529T^{2} $
	29	$ 1 + 34.5iT - 841T^{2} $
	31	$ 1 - 24.7T + 961T^{2} $
	37	$ 1 + 1.36T + 1.36e3T^{2} $
	41	$ 1 + 50.9iT - 1.68e3T^{2} $
	43	$ 1 - 62.8T + 1.84e3T^{2} $
	47	$ 1 - 23.5iT - 2.20e3T^{2} $
	53	$ 1 + 4.32iT - 2.80e3T^{2} $
	59	$ 1 + 69.2iT - 3.48e3T^{2} $
	61	$ 1 + 36.9T + 3.72e3T^{2} $
	67	$ 1 + 9.12T + 4.48e3T^{2} $
	71	$ 1 + 33.2iT - 5.04e3T^{2} $
	73	$ 1 + 8.76T + 5.32e3T^{2} $
	79	$ 1 + 3.01T + 6.24e3T^{2} $
	83	$ 1 + 43.9iT - 6.88e3T^{2} $
	89	$ 1 + 105. iT - 7.92e3T^{2} $
	97	$ 1 - 102.T + 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

−9.713605764365495964088721005914, −9.285226739732629387058491722611, −7.60981911397142270148640028134, −7.05067719573992070744170274702, −5.85673269928520070977571078377, −4.69886566330150974334129433235, −3.48121381102572871161516502510, −2.95727290582062952216482524724, −2.23345313990705584542537128422, −0.46652809764785180634861064418, 0.70198565184991028268056934745, 2.87541305349938153512127034466, 4.37719813385210866804740822914, 5.00770489036060016399701680442, 5.74148302220411552187854893645, 6.70763369001322924531814379214, 7.35609616547162077421676414163, 8.159084460558733803321492936702, 9.082501232038950799782450894220, 9.489464300803367062200414801291

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.9451036990\)
\(L(\frac12)\)	\(\approx\)	\(0.9451036990\)
\(L(2)\)		not available
\(L(1)\)		not available

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