LMFDB - L-function 2-48e2-24.11-c1-0-12

Dirichlet series

L(s) = 1

− 3.41·5^-s + 4.82i·7^-s − 2.82i·11^-s − 2.82i·13^-s − 5.41i·17^-s − 5.65·19^-s − 1.17·23^-s + 6.65·25^-s + 0.585·29^-s + 3.17i·31^-s − 16.4i·35^-s + 3.65i·37^-s + 2.58i·41^-s + 9.65·43^-s + 12.4·47^-s + ⋯

L(s) = 1

− 1.52·5^-s + 1.82i·7^-s − 0.852i·11^-s − 0.784i·13^-s − 1.31i·17^-s − 1.29·19^-s − 0.244·23^-s + 1.33·25^-s + 0.108·29^-s + 0.569i·31^-s − 2.78i·35^-s + 0.601i·37^-s + 0.403i·41^-s + 1.47·43^-s + 1.82·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2304$ = $2^{8} \cdot 3^{2}$
Sign:	$0.985 + 0.169i$
Analytic conductor:	$18.3975$
Root analytic conductor:	$4.28923$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2304} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2304,\ (\ :1/2),\ 0.985 + 0.169i)$

Particular Values

$L(1)$	$\approx$	$0.9860913608$
$L(\frac12)$	$\approx$	$0.9860913608$
$L(\frac{3}{2})$		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 $
good	5	$ 1 + 3.41T + 5T^{2} $
	7	$ 1 - 4.82iT - 7T^{2} $
	11	$ 1 + 2.82iT - 11T^{2} $
	13	$ 1 + 2.82iT - 13T^{2} $
	17	$ 1 + 5.41iT - 17T^{2} $
	19	$ 1 + 5.65T + 19T^{2} $
	23	$ 1 + 1.17T + 23T^{2} $
	29	$ 1 - 0.585T + 29T^{2} $
	31	$ 1 - 3.17iT - 31T^{2} $
	37	$ 1 - 3.65iT - 37T^{2} $
	41	$ 1 - 2.58iT - 41T^{2} $
	43	$ 1 - 9.65T + 43T^{2} $
	47	$ 1 - 12.4T + 47T^{2} $
	53	$ 1 - 5.07T + 53T^{2} $
	59	$ 1 - 2.34iT - 59T^{2} $
	61	$ 1 + 7.65iT - 61T^{2} $
	67	$ 1 - 12T + 67T^{2} $
	71	$ 1 - 4.48T + 71T^{2} $
	73	$ 1 + 4T + 73T^{2} $
	79	$ 1 + 6.48iT - 79T^{2} $
	83	$ 1 - 5.17iT - 83T^{2} $
	89	$ 1 - 12.2iT - 89T^{2} $
	97	$ 1 - 13.6T + 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

−8.742628929425123701147954041755, −8.333038287695827614147577856186, −7.65147872539780213934625132532, −6.65233453745634339572586103720, −5.76467461938641941800474404669, −5.07595883104042653640973737903, −4.08787944716623970609163785502, −3.08439136072330616554662316292, −2.45059733443672442533801330584, −0.54037091130049667997643711500, 0.72576440225218819714584265843, 2.11233070842836190345554822279, 3.80644215437550200190973260076, 4.05692503230653688573751212926, 4.54151946418519482345212078162, 6.08787232981047724639073416754, 7.06817821417529441957166433835, 7.36292877383555365793892919900, 8.111391113212410065979088344204, 8.849828730927724023909351774414

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9860913608\)
\(L(\frac12)\)	\(\approx\)	\(0.9860913608\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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