LMFDB - L-function 2-2736-19.18-c2-0-76

Dirichlet series

L(s) = 1

+ 5.08·5^-s + 11.6·7^-s − 14.6·11^-s − 16.9i·13^-s − 8.79·17^-s + (11.9 + 14.7i)19^-s − 6.84·23^-s + 0.900·25^-s + 45.8i·29^-s − 50.1i·31^-s + 59.5·35^-s − 35.1i·37^-s − 60.4i·41^-s + 75.0·43^-s + 52.6·47^-s + ⋯

L(s) = 1

+ 1.01·5^-s + 1.67·7^-s − 1.33·11^-s − 1.30i·13^-s − 0.517·17^-s + (0.629 + 0.776i)19^-s − 0.297·23^-s + 0.0360·25^-s + 1.57i·29^-s − 1.61i·31^-s + 1.70·35^-s − 0.951i·37^-s − 1.47i·41^-s + 1.74·43^-s + 1.12·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$0.629 + 0.776i$
Analytic conductor:	$74.5506$
Root analytic conductor:	$8.63426$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (721, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1),\ 0.629 + 0.776i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.882199830$
$L(\frac12)$	$\approx$	$2.882199830$
$L(2)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	2	$ 1 $
	3	$ 1 $
	19	$ 1 + (-11.9 - 14.7i)T $
good	5	$ 1 - 5.08T + 25T^{2} $
	7	$ 1 - 11.6T + 49T^{2} $
	11	$ 1 + 14.6T + 121T^{2} $
	13	$ 1 + 16.9iT - 169T^{2} $
	17	$ 1 + 8.79T + 289T^{2} $
	23	$ 1 + 6.84T + 529T^{2} $
	29	$ 1 - 45.8iT - 841T^{2} $
	31	$ 1 + 50.1iT - 961T^{2} $
	37	$ 1 + 35.1iT - 1.36e3T^{2} $
	41	$ 1 + 60.4iT - 1.68e3T^{2} $
	43	$ 1 - 75.0T + 1.84e3T^{2} $
	47	$ 1 - 52.6T + 2.20e3T^{2} $
	53	$ 1 + 93.2iT - 2.80e3T^{2} $
	59	$ 1 + 13.0iT - 3.48e3T^{2} $
	61	$ 1 + 1.36T + 3.72e3T^{2} $
	67	$ 1 + 69.4iT - 4.48e3T^{2} $
	71	$ 1 - 58.7iT - 5.04e3T^{2} $
	73	$ 1 + 41.4T + 5.32e3T^{2} $
	79	$ 1 - 90.3iT - 6.24e3T^{2} $
	83	$ 1 - 155.T + 6.88e3T^{2} $
	89	$ 1 + 73.4iT - 7.92e3T^{2} $
	97	$ 1 + 36.7iT - 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

−8.421323481944750505491706529625, −7.77683667437242551432501613949, −7.32656353730207363907028204937, −5.79505708533325690784160292485, −5.54873995483561408303174846132, −4.89071877417363341649582125012, −3.75642839934001563916145633742, −2.45131766860232634147348981832, −1.93247136668401442405934993770, −0.67604616745932114795265115783, 1.15845264224576012330772915674, 2.08634751521741718956412164686, 2.68634674499466023086899597612, 4.36241440676387007366510026718, 4.80221726600205177723528489426, 5.59973723864994364255238618793, 6.36431061612603040345180263196, 7.41492081232891331009112815190, 7.923506825417813513358177606205, 8.835056838938740352543029183103

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.882199830\)
\(L(\frac12)\)	\(\approx\)	\(2.882199830\)
\(L(2)\)		not available
\(L(1)\)		not available

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