LMFDB - L-function 2-33e2-3.2-c2-0-27

Dirichlet series

L(s) = 1

− 2.80i·2^-s − 3.87·4^-s + 2.03i·5^-s − 0.504·7^-s − 0.356i·8^-s + 5.70·10^-s + 22.0·13^-s + 1.41i·14^-s − 16.4·16^-s + 11.9i·17^-s − 5.92·19^-s − 7.86i·20^-s + 42.7i·23^-s + 20.8·25^-s − 61.9i·26^-s + ⋯

L(s) = 1

− 1.40i·2^-s − 0.968·4^-s + 0.406i·5^-s − 0.0720·7^-s − 0.0445i·8^-s + 0.570·10^-s + 1.69·13^-s + 0.101i·14^-s − 1.03·16^-s + 0.704i·17^-s − 0.311·19^-s − 0.393i·20^-s + 1.85i·23^-s + 0.834·25^-s − 2.38i·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

$L(\frac{3}{2})$	$\approx$	$1.930330095$
$L(\frac12)$	$\approx$	$1.930330095$
$L(2)$		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 + 2.80iT - 4T^{2} $
	5	$ 1 - 2.03iT - 25T^{2} $
	7	$ 1 + 0.504T + 49T^{2} $
	13	$ 1 - 22.0T + 169T^{2} $
	17	$ 1 - 11.9iT - 289T^{2} $
	19	$ 1 + 5.92T + 361T^{2} $
	23	$ 1 - 42.7iT - 529T^{2} $
	29	$ 1 - 36.4iT - 841T^{2} $
	31	$ 1 + 3.49T + 961T^{2} $
	37	$ 1 + 32.8T + 1.36e3T^{2} $
	41	$ 1 - 27.2iT - 1.68e3T^{2} $
	43	$ 1 - 58.0T + 1.84e3T^{2} $
	47	$ 1 - 11.8iT - 2.20e3T^{2} $
	53	$ 1 + 61.7iT - 2.80e3T^{2} $
	59	$ 1 + 59.9iT - 3.48e3T^{2} $
	61	$ 1 - 49.5T + 3.72e3T^{2} $
	67	$ 1 - 108.T + 4.48e3T^{2} $
	71	$ 1 + 51.4iT - 5.04e3T^{2} $
	73	$ 1 - 123.T + 5.32e3T^{2} $
	79	$ 1 - 9.12T + 6.24e3T^{2} $
	83	$ 1 + 88.2iT - 6.88e3T^{2} $
	89	$ 1 - 4.52iT - 7.92e3T^{2} $
	97	$ 1 - 71.1T + 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

−9.731135062729649868763304717493, −8.993729722474859411090233341290, −8.170236012466068510398913138979, −6.94990787589120012217214412360, −6.18521053390602217384830818240, −5.01359089791674693840793189009, −3.57871185876543514126968879247, −3.46544409900859677498883544901, −1.98477896599178884780928844724, −1.09465044390140256520129479410, 0.71287368430350526799616229463, 2.43853256572762019495544651512, 3.95700598650216172299074139179, 4.82638159116641483917915353404, 5.77922629242179117187116977317, 6.44587428026018575114057930954, 7.15021175914762608884891873403, 8.293710881287694166082779462831, 8.569334262125757888461615742699, 9.407533753888807838033275354215

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.930330095\)
\(L(\frac12)\)	\(\approx\)	\(1.930330095\)
\(L(2)\)		not available
\(L(1)\)		not available

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