LMFDB - L-function 2-12e3-12.11-c1-0-13

Dirichlet series

L(s) = 1

− 3i·5^-s + 1.73i·7^-s + 5.19·11^-s + 2·13^-s + 6i·17^-s + 6.92i·19^-s − 4·25^-s − 6i·29^-s + 5.19i·31^-s + 5.19·35^-s − 8·37^-s − 10.3i·43^-s + 10.3·47^-s + 4·49^-s + 9i·53^-s + ⋯

L(s) = 1

− 1.34i·5^-s + 0.654i·7^-s + 1.56·11^-s + 0.554·13^-s + 1.45i·17^-s + 1.58i·19^-s − 0.800·25^-s − 1.11i·29^-s + 0.933i·31^-s + 0.878·35^-s − 1.31·37^-s − 1.58i·43^-s + 1.51·47^-s + 0.571·49^-s + 1.23i·53^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$1$
Analytic conductor:	$13.7981$
Root analytic conductor:	$3.71458$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (1727, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :1/2),\ 1)$

Particular Values

$L(1)$	$\approx$	$1.934827741$
$L(\frac12)$	$\approx$	$1.934827741$
$L(\frac{3}{2})$		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 + 3iT - 5T^{2} $
	7	$ 1 - 1.73iT - 7T^{2} $
	11	$ 1 - 5.19T + 11T^{2} $
	13	$ 1 - 2T + 13T^{2} $
	17	$ 1 - 6iT - 17T^{2} $
	19	$ 1 - 6.92iT - 19T^{2} $
	23	$ 1 + 23T^{2} $
	29	$ 1 + 6iT - 29T^{2} $
	31	$ 1 - 5.19iT - 31T^{2} $
	37	$ 1 + 8T + 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 + 10.3iT - 43T^{2} $
	47	$ 1 - 10.3T + 47T^{2} $
	53	$ 1 - 9iT - 53T^{2} $
	59	$ 1 - 10.3T + 59T^{2} $
	61	$ 1 - 4T + 61T^{2} $
	67	$ 1 + 3.46iT - 67T^{2} $
	71	$ 1 - 10.3T + 71T^{2} $
	73	$ 1 - T + 73T^{2} $
	79	$ 1 - 3.46iT - 79T^{2} $
	83	$ 1 + 5.19T + 83T^{2} $
	89	$ 1 + 6iT - 89T^{2} $
	97	$ 1 + 5T + 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

−9.005599075291961197632721012947, −8.693180799661226862809901034165, −8.070740179229920224203064774766, −6.80946026581393295350425941531, −5.90618445745239695545119098405, −5.43593328386143964436174316009, −4.09308215720697952450186769509, −3.77796534111514868149395069128, −1.93600067004104086248348938736, −1.16012666804914889793893556240, 0.919011476211340647445773075409, 2.45104155301302268322869647163, 3.37529558553623354544624888830, 4.13947901801013514964300761021, 5.23580117772196954355247479927, 6.48606728258172651779128529516, 6.90647291291446221822881990012, 7.37544013035595651079439416360, 8.681129711471335363924712308743, 9.349430018754690892627097515960

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.934827741\)
\(L(\frac12)\)	\(\approx\)	\(1.934827741\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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