LMFDB - L-function 2-2736-57.56-c1-0-16

Dirichlet series

L(s) = 1

+ 2.32i·5^-s − 4.34·7^-s + 0.264i·11^-s − 6.38i·13^-s − 2.56i·17^-s + (−2.62 + 3.47i)19^-s + 2.66i·23^-s − 0.426·25^-s + 8.92·29^-s − 1.78i·31^-s − 10.1i·35^-s + 3.08i·37^-s − 7.44·41^-s + 5.05·43^-s − 7.45i·47^-s + ⋯

L(s) = 1

+ 1.04i·5^-s − 1.64·7^-s + 0.0797i·11^-s − 1.76i·13^-s − 0.621i·17^-s + (−0.602 + 0.798i)19^-s + 0.556i·23^-s − 0.0852·25^-s + 1.65·29^-s − 0.320i·31^-s − 1.71i·35^-s + 0.507i·37^-s − 1.16·41^-s + 0.770·43^-s − 1.08i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0310i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$0.999 + 0.0310i$
Analytic conductor:	$21.8470$
Root analytic conductor:	$4.67408$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (1025, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1/2),\ 0.999 + 0.0310i)$

Particular Values

$L(1)$	$\approx$	$1.222389101$
$L(\frac12)$	$\approx$	$1.222389101$
$L(\frac{3}{2})$		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 + (2.62 - 3.47i)T $
good	5	$ 1 - 2.32iT - 5T^{2} $
	7	$ 1 + 4.34T + 7T^{2} $
	11	$ 1 - 0.264iT - 11T^{2} $
	13	$ 1 + 6.38iT - 13T^{2} $
	17	$ 1 + 2.56iT - 17T^{2} $
	23	$ 1 - 2.66iT - 23T^{2} $
	29	$ 1 - 8.92T + 29T^{2} $
	31	$ 1 + 1.78iT - 31T^{2} $
	37	$ 1 - 3.08iT - 37T^{2} $
	41	$ 1 + 7.44T + 41T^{2} $
	43	$ 1 - 5.05T + 43T^{2} $
	47	$ 1 + 7.45iT - 47T^{2} $
	53	$ 1 - 4.10T + 53T^{2} $
	59	$ 1 - 8.36T + 59T^{2} $
	61	$ 1 + 8.45T + 61T^{2} $
	67	$ 1 + 2.35iT - 67T^{2} $
	71	$ 1 + 5.02T + 71T^{2} $
	73	$ 1 - 14.9T + 73T^{2} $
	79	$ 1 - 6.23iT - 79T^{2} $
	83	$ 1 + 16.0iT - 83T^{2} $
	89	$ 1 - 8.25T + 89T^{2} $
	97	$ 1 - 15.1iT - 97T^{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

−8.835889956066646458029244942637, −7.978354811411018515834225495700, −7.18887237004807573608540638486, −6.50424276167379447054815611635, −5.97602609581574671597572386515, −5.04081282328611217586663167648, −3.65114503787719019198442212211, −3.16240661415087748816052868353, −2.46686424274784030374222506856, −0.59174316994332836852671077856, 0.75219107865496680305110247954, 2.10382009832878025367634973524, 3.16589957872834171224732966038, 4.22574265015476693678313698671, 4.70291604758952722776352891955, 5.90956205910014349269203132808, 6.59899419128853149729351030964, 7.02448222631768992887590140276, 8.403880382401392342207228820220, 8.867792564601474528985838759062

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.222389101\)
\(L(\frac12)\)	\(\approx\)	\(1.222389101\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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