LMFDB - L-function 2-5760-24.11-c1-0-57

Dirichlet series

L(s) = 1

− 5^-s − 0.116i·7^-s − 5.10i·11^-s − 3.81i·13^-s − 2.16i·17^-s + 0.828·19^-s + 2.39·23^-s + 25^-s − 5.06·29^-s + 1.83i·31^-s + 0.116i·35^-s − 0.421i·37^-s − 4.80i·41^-s − 7.39·43^-s − 3.88·47^-s + ⋯

L(s) = 1

− 0.447·5^-s − 0.0440i·7^-s − 1.54i·11^-s − 1.05i·13^-s − 0.525i·17^-s + 0.190·19^-s + 0.499·23^-s + 0.200·25^-s − 0.939·29^-s + 0.329i·31^-s + 0.0196i·35^-s − 0.0692i·37^-s − 0.750i·41^-s − 1.12·43^-s − 0.567·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5760$ = $2^{7} \cdot 3^{2} \cdot 5$
Sign:	$-0.985 + 0.169i$
Analytic conductor:	$45.9938$
Root analytic conductor:	$6.78187$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5760} (4031, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5760,\ (\ :1/2),\ -0.985 + 0.169i)$

Particular Values

$L(1)$	$\approx$	$0.7859935153$
$L(\frac12)$	$\approx$	$0.7859935153$
$L(\frac{3}{2})$		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 $
	5	$ 1 + T $
good	7	$ 1 + 0.116iT - 7T^{2} $
	11	$ 1 + 5.10iT - 11T^{2} $
	13	$ 1 + 3.81iT - 13T^{2} $
	17	$ 1 + 2.16iT - 17T^{2} $
	19	$ 1 - 0.828T + 19T^{2} $
	23	$ 1 - 2.39T + 23T^{2} $
	29	$ 1 + 5.06T + 29T^{2} $
	31	$ 1 - 1.83iT - 31T^{2} $
	37	$ 1 + 0.421iT - 37T^{2} $
	41	$ 1 + 4.80iT - 41T^{2} $
	43	$ 1 + 7.39T + 43T^{2} $
	47	$ 1 + 3.88T + 47T^{2} $
	53	$ 1 - 7.39T + 53T^{2} $
	59	$ 1 + 7.60iT - 59T^{2} $
	61	$ 1 - 1.17iT - 61T^{2} $
	67	$ 1 - 3.39T + 67T^{2} $
	71	$ 1 + 13.0T + 71T^{2} $
	73	$ 1 + 7.65T + 73T^{2} $
	79	$ 1 - 7.49iT - 79T^{2} $
	83	$ 1 + 1.09iT - 83T^{2} $
	89	$ 1 - 15.6iT - 89T^{2} $
	97	$ 1 + 4.06T + 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

−7.78391711642023509613097459447, −7.18715738224646120720517522045, −6.33017264621681236500310297720, −5.51589718389375422664969875839, −5.08706026699377368898593652615, −3.87396678954188216868415586324, −3.32761575058947119148011066637, −2.57386900591019539441423804627, −1.13413327915060873349981294316, −0.21892304523687679296731591525, 1.45421150391781325594868390018, 2.19278820064574014103375060601, 3.28180869456361667953334963611, 4.23889456940082067184280152168, 4.60645814886477070042853456548, 5.56254892446515200458911732041, 6.44203439871789757920157012004, 7.21584831045744453685822899603, 7.49012878729655714447900218126, 8.528747020444721469325435817460

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7859935153\)
\(L(\frac12)\)	\(\approx\)	\(0.7859935153\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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