LMFDB - L-function 2-5760-12.11-c1-0-10

Dirichlet series

L(s) = 1

− i·5^-s − 1.41i·7^-s − 3.55·11^-s − 6.57·13^-s − 0.138i·17^-s + 4.96i·19^-s + 3.16·23^-s − 25^-s − 2.19i·29^-s − 9.16i·31^-s − 1.41·35^-s + 0.723·37^-s + 1.21i·41^-s − 7.30i·43^-s + 0.494·47^-s + ⋯

L(s) = 1

− 0.447i·5^-s − 0.534i·7^-s − 1.07·11^-s − 1.82·13^-s − 0.0335i·17^-s + 1.13i·19^-s + 0.659·23^-s − 0.200·25^-s − 0.407i·29^-s − 1.64i·31^-s − 0.239·35^-s + 0.119·37^-s + 0.190i·41^-s − 1.11i·43^-s + 0.0721·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5760$ = $2^{7} \cdot 3^{2} \cdot 5$
Sign:	$0.577 - 0.816i$
Analytic conductor:	$45.9938$
Root analytic conductor:	$6.78187$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5760} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5760,\ (\ :1/2),\ 0.577 - 0.816i)$

Particular Values

$L(1)$	$\approx$	$0.9234212879$
$L(\frac12)$	$\approx$	$0.9234212879$
$L(\frac{3}{2})$		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 + iT $
good	7	$ 1 + 1.41iT - 7T^{2} $
	11	$ 1 + 3.55T + 11T^{2} $
	13	$ 1 + 6.57T + 13T^{2} $
	17	$ 1 + 0.138iT - 17T^{2} $
	19	$ 1 - 4.96iT - 19T^{2} $
	23	$ 1 - 3.16T + 23T^{2} $
	29	$ 1 + 2.19iT - 29T^{2} $
	31	$ 1 + 9.16iT - 31T^{2} $
	37	$ 1 - 0.723T + 37T^{2} $
	41	$ 1 - 1.21iT - 41T^{2} $
	43	$ 1 + 7.30iT - 43T^{2} $
	47	$ 1 - 0.494T + 47T^{2} $
	53	$ 1 - 12.6iT - 53T^{2} $
	59	$ 1 + 6.29T + 59T^{2} $
	61	$ 1 - 13.4T + 61T^{2} $
	67	$ 1 - 13.8iT - 67T^{2} $
	71	$ 1 + 8.47T + 71T^{2} $
	73	$ 1 + 5.10T + 73T^{2} $
	79	$ 1 - 6.81iT - 79T^{2} $
	83	$ 1 + 4.48T + 83T^{2} $
	89	$ 1 - 11.2iT - 89T^{2} $
	97	$ 1 - 9.30T + 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

−8.038990557497023422681223341026, −7.53177003045404514727312473986, −7.07549717177254816322842629047, −5.88036759962422843625454878364, −5.37779094754097360620151766077, −4.56774674024260410113183399633, −3.96155040485297058483792551393, −2.77376550792941511019264777793, −2.14489580923368089287263154493, −0.792784604787799776695625534659, 0.30113037241840129123101839012, 1.94519060470330701265896931466, 2.75252893529813862300864346673, 3.19323913800314850369120655051, 4.69925944039843637634902398108, 4.97973833129552215924561909224, 5.75854805879868079226543984738, 6.84750744378620815529412496540, 7.17097021436274615329483027325, 7.935041674956165888590019859646

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9234212879\)
\(L(\frac12)\)	\(\approx\)	\(0.9234212879\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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