LMFDB - L-function 2-48e2-24.11-c1-0-26

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

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

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

Invariants

Degree:	$2$
Conductor:	$2304$ = $2^{8} \cdot 3^{2}$
Sign:	$0.169 + 0.985i$
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:	$(2,\ 2304,\ (\ :1/2),\ 0.169 + 0.985i)$

Particular Values

$L(1)$	$\approx$	$2.380635137$
$L(\frac12)$	$\approx$	$2.380635137$
$L(\frac{3}{2})$		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

−8.894288836591913855677111017774, −7.983687297878934146064798741805, −7.31770245710811620837178085465, −6.34705804031500724659682194126, −5.82469466625548586968915220488, −4.95590953664081951571103748715, −3.83748474723901262805046475903, −3.09378886361479788636458209660, −1.70625626132795416293482729059, −0.826585340299214730326717437238, 1.61123607492311535459949866335, 2.32251120395997807349617784552, 3.05166164318882793353193493911, 4.73052820495714928541401268023, 5.32822658451853229592401122351, 5.91608311452727762746857227518, 6.72406615080626481217633553512, 7.54725142908868946164707095591, 8.792287002570371706246089758869, 9.277155654074965225491697640459

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.380635137\)
\(L(\frac12)\)	\(\approx\)	\(2.380635137\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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