LMFDB - L-function 2-2736-12.11-c1-0-5

Dirichlet series

L(s) = 1

+ 2.95i·5^-s − 0.569i·7^-s − 1.70·11^-s + 3.55·13^-s + 4.91i·17^-s − i·19^-s − 8.27·23^-s − 3.75·25^-s + 2.39i·29^-s + 4.25i·31^-s + 1.68·35^-s − 5.95·37^-s − 3.11i·41^-s + 3.81i·43^-s + 5.85·47^-s + ⋯

L(s) = 1

+ 1.32i·5^-s − 0.215i·7^-s − 0.515·11^-s + 0.985·13^-s + 1.19i·17^-s − 0.229i·19^-s − 1.72·23^-s − 0.750·25^-s + 0.444i·29^-s + 0.764i·31^-s + 0.285·35^-s − 0.978·37^-s − 0.486i·41^-s + 0.582i·43^-s + 0.854·47^-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

$L(1)$	$\approx$	$1.013180047$
$L(\frac12)$	$\approx$	$1.013180047$
$L(\frac{3}{2})$		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 - 2.95iT - 5T^{2} $
	7	$ 1 + 0.569iT - 7T^{2} $
	11	$ 1 + 1.70T + 11T^{2} $
	13	$ 1 - 3.55T + 13T^{2} $
	17	$ 1 - 4.91iT - 17T^{2} $
	23	$ 1 + 8.27T + 23T^{2} $
	29	$ 1 - 2.39iT - 29T^{2} $
	31	$ 1 - 4.25iT - 31T^{2} $
	37	$ 1 + 5.95T + 37T^{2} $
	41	$ 1 + 3.11iT - 41T^{2} $
	43	$ 1 - 3.81iT - 43T^{2} $
	47	$ 1 - 5.85T + 47T^{2} $
	53	$ 1 - 4.30iT - 53T^{2} $
	59	$ 1 + 2.13T + 59T^{2} $
	61	$ 1 - 4.19T + 61T^{2} $
	67	$ 1 + 12.3iT - 67T^{2} $
	71	$ 1 + 12.7T + 71T^{2} $
	73	$ 1 + 10.8T + 73T^{2} $
	79	$ 1 - 2.29iT - 79T^{2} $
	83	$ 1 + 1.73T + 83T^{2} $
	89	$ 1 - 14.6iT - 89T^{2} $
	97	$ 1 + 5.53T + 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

−9.077379438895005084755051204898, −8.305216000954906143598332386636, −7.61973670442856400759772867148, −6.81456633251776210537366335873, −6.18227254728805817250694070930, −5.50209466253421810442448800621, −4.17054337644335421958018519784, −3.54788811499127649144203509055, −2.63800143001434969722709232814, −1.57814163014678059619296878724, 0.32132383816241762472332224785, 1.51410251839364476508003036059, 2.60765038900789598834287051506, 3.87331095301744723791755516782, 4.52744957078661671118262542303, 5.51995708821947778924383566667, 5.88007942882765373916819279389, 7.07823325699509063153932765509, 7.934718189987379615395414884386, 8.538901874094857999640474499491

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.013180047\)
\(L(\frac12)\)	\(\approx\)	\(1.013180047\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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