LMFDB - L-function 2-2736-12.11-c1-0-23

Dirichlet series

L(s) = 1

+ 2.62i·5^-s − 0.815i·7^-s + 2.98·11^-s − 1.44·13^-s − 8.05i·17^-s + i·19^-s + 2.16·23^-s − 1.89·25^-s − 9.88i·29^-s − 10.1i·31^-s + 2.14·35^-s − 7.24·37^-s + 3.87i·41^-s + 11.2i·43^-s + 12.5·47^-s + ⋯

L(s) = 1

+ 1.17i·5^-s − 0.308i·7^-s + 0.901·11^-s − 0.400·13^-s − 1.95i·17^-s + 0.229i·19^-s + 0.450·23^-s − 0.379·25^-s − 1.83i·29^-s − 1.83i·31^-s + 0.361·35^-s − 1.19·37^-s + 0.605i·41^-s + 1.71i·43^-s + 1.83·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2736$ = $2^{4} \cdot 3^{2} \cdot 19$
Sign:	$0.908 + 0.418i$
Analytic conductor:	$21.8470$
Root analytic conductor:	$4.67408$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2736} (2015, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2736,\ (\ :1/2),\ 0.908 + 0.418i)$

Particular Values

$L(1)$	$\approx$	$1.761607002$
$L(\frac12)$	$\approx$	$1.761607002$
$L(\frac{3}{2})$		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 - iT $
good	5	$ 1 - 2.62iT - 5T^{2} $
	7	$ 1 + 0.815iT - 7T^{2} $
	11	$ 1 - 2.98T + 11T^{2} $
	13	$ 1 + 1.44T + 13T^{2} $
	17	$ 1 + 8.05iT - 17T^{2} $
	23	$ 1 - 2.16T + 23T^{2} $
	29	$ 1 + 9.88iT - 29T^{2} $
	31	$ 1 + 10.1iT - 31T^{2} $
	37	$ 1 + 7.24T + 37T^{2} $
	41	$ 1 - 3.87iT - 41T^{2} $
	43	$ 1 - 11.2iT - 43T^{2} $
	47	$ 1 - 12.5T + 47T^{2} $
	53	$ 1 + 8.04iT - 53T^{2} $
	59	$ 1 + 9.48T + 59T^{2} $
	61	$ 1 - 4.22T + 61T^{2} $
	67	$ 1 - 11.4iT - 67T^{2} $
	71	$ 1 + 4.11T + 71T^{2} $
	73	$ 1 - 12.1T + 73T^{2} $
	79	$ 1 + 15.4iT - 79T^{2} $
	83	$ 1 + 0.745T + 83T^{2} $
	89	$ 1 + 6.82iT - 89T^{2} $
	97	$ 1 + 0.975T + 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

−8.914142638524179608758087810989, −7.68899040643597537685446063607, −7.32102872171368568314935882781, −6.55605846702199829570508453165, −5.88216372759936818820745068430, −4.75207016491370065788418646695, −3.95403976756810728825249602107, −2.97039114771231468381288853576, −2.26753154259086900556960266731, −0.66175930849464755318346551190, 1.11338478379528762336503230582, 1.90696885454559848550403061806, 3.37149739611038553537338944255, 4.13320029812814910894986837816, 5.07736422048519314445641064763, 5.60352934988165922982185719671, 6.65045331655931301860198415939, 7.28752733013918097822690360153, 8.547159096721573871906048397252, 8.748115123058955538541531974701

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.761607002\)
\(L(\frac12)\)	\(\approx\)	\(1.761607002\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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