LMFDB - L-function 2-48e2-24.11-c1-0-18

Dirichlet series

L(s) = 1

+ 0.585·5^-s − 0.828i·7^-s − 2.82i·11^-s + 2.82i·13^-s + 2.58i·17^-s + 5.65·19^-s + 6.82·23^-s − 4.65·25^-s − 3.41·29^-s + 8.82i·31^-s − 0.485i·35^-s − 7.65i·37^-s − 5.41i·41^-s − 1.65·43^-s + 4.48·47^-s + ⋯

L(s) = 1

+ 0.261·5^-s − 0.313i·7^-s − 0.852i·11^-s + 0.784i·13^-s + 0.627i·17^-s + 1.29·19^-s + 1.42·23^-s − 0.931·25^-s − 0.634·29^-s + 1.58i·31^-s − 0.0820i·35^-s − 1.25i·37^-s − 0.845i·41^-s − 0.252·43^-s + 0.654·47^-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

$L(1)$	$\approx$	$1.938148554$
$L(\frac12)$	$\approx$	$1.938148554$
$L(\frac{3}{2})$		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 - 0.585T + 5T^{2} $
	7	$ 1 + 0.828iT - 7T^{2} $
	11	$ 1 + 2.82iT - 11T^{2} $
	13	$ 1 - 2.82iT - 13T^{2} $
	17	$ 1 - 2.58iT - 17T^{2} $
	19	$ 1 - 5.65T + 19T^{2} $
	23	$ 1 - 6.82T + 23T^{2} $
	29	$ 1 + 3.41T + 29T^{2} $
	31	$ 1 - 8.82iT - 31T^{2} $
	37	$ 1 + 7.65iT - 37T^{2} $
	41	$ 1 + 5.41iT - 41T^{2} $
	43	$ 1 + 1.65T + 43T^{2} $
	47	$ 1 - 4.48T + 47T^{2} $
	53	$ 1 - 9.07T + 53T^{2} $
	59	$ 1 + 13.6iT - 59T^{2} $
	61	$ 1 - 3.65iT - 61T^{2} $
	67	$ 1 - 12T + 67T^{2} $
	71	$ 1 - 12.4T + 71T^{2} $
	73	$ 1 + 4T + 73T^{2} $
	79	$ 1 - 10.4iT - 79T^{2} $
	83	$ 1 + 10.8iT - 83T^{2} $
	89	$ 1 + 3.75iT - 89T^{2} $
	97	$ 1 - 2.34T + 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

−9.029422085606994773386734505795, −8.311447756277913781804691326997, −7.29911654830069432005677844506, −6.80472871834095659322293994314, −5.71092882633406978161809065941, −5.20815156810270848403629058594, −3.97194765791397514818502033809, −3.32120340323026275174472131742, −2.07904868722621480438841046470, −0.904655123399420313629413041588, 0.954886663720236993772733085453, 2.28848788511216843242081316202, 3.11341700999844826315095394977, 4.22587661591381140889934911571, 5.24097923808208837279425431541, 5.68337696325808554607420135233, 6.82681156278023573655577692159, 7.48153331845668943046584061043, 8.160224414792089058577790133349, 9.266734984379155764704396733817

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.938148554\)
\(L(\frac12)\)	\(\approx\)	\(1.938148554\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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