LMFDB - L-function 2-12e3-12.11-c3-0-70

Dirichlet series

L(s) = 1

− 3.33i·5^-s − 16.9i·7^-s + 16.8·11^-s − 25.0·13^-s − 116. i·17^-s + 85.4i·19^-s + 158.·23^-s + 113.·25^-s + 269. i·29^-s + 36.0i·31^-s − 56.3·35^-s + 353.·37^-s − 144. i·41^-s − 368. i·43^-s − 397.·47^-s + ⋯

L(s) = 1

− 0.297i·5^-s − 0.912i·7^-s + 0.461·11^-s − 0.535·13^-s − 1.66i·17^-s + 1.03i·19^-s + 1.43·23^-s + 0.911·25^-s + 1.72i·29^-s + 0.209i·31^-s − 0.271·35^-s + 1.57·37^-s − 0.550i·41^-s − 1.30i·43^-s − 1.23·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1728$ = $2^{6} \cdot 3^{3}$
Sign:	$i$
Analytic conductor:	$101.955$
Root analytic conductor:	$10.0972$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1728} (1727, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1728,\ (\ :3/2),\ i)$

Particular Values

$L(2)$	$\approx$	$2.062634000$
$L(\frac12)$	$\approx$	$2.062634000$
$L(\frac{5}{2})$		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 + 3.33iT - 125T^{2} $
	7	$ 1 + 16.9iT - 343T^{2} $
	11	$ 1 - 16.8T + 1.33e3T^{2} $
	13	$ 1 + 25.0T + 2.19e3T^{2} $
	17	$ 1 + 116. iT - 4.91e3T^{2} $
	19	$ 1 - 85.4iT - 6.85e3T^{2} $
	23	$ 1 - 158.T + 1.21e4T^{2} $
	29	$ 1 - 269. iT - 2.43e4T^{2} $
	31	$ 1 - 36.0iT - 2.97e4T^{2} $
	37	$ 1 - 353.T + 5.06e4T^{2} $
	41	$ 1 + 144. iT - 6.89e4T^{2} $
	43	$ 1 + 368. iT - 7.95e4T^{2} $
	47	$ 1 + 397.T + 1.03e5T^{2} $
	53	$ 1 + 96.0iT - 1.48e5T^{2} $
	59	$ 1 + 294.T + 2.05e5T^{2} $
	61	$ 1 + 146.T + 2.26e5T^{2} $
	67	$ 1 + 301. iT - 3.00e5T^{2} $
	71	$ 1 - 687.T + 3.57e5T^{2} $
	73	$ 1 + 312.T + 3.89e5T^{2} $
	79	$ 1 + 602. iT - 4.93e5T^{2} $
	83	$ 1 - 1.33e3T + 5.71e5T^{2} $
	89	$ 1 - 856. iT - 7.04e5T^{2} $
	97	$ 1 - 218.T + 9.12e5T^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−8.968266614900514363159072769555, −7.83760345598200184768420047189, −7.14225567990881873331049982479, −6.60506687532158659379809125445, −5.23754675000018181148442996320, −4.78071025153029796086473346982, −3.67793239412216490990103306169, −2.81319344466420141548254333628, −1.38415736317922311881990103554, −0.52512812925074269123484935662, 1.02324376577340074732337762254, 2.31426223490808048615016910313, 3.02745050024676997616682121005, 4.25230553500349154517007582318, 5.03570239175114490488562624627, 6.13358931116120451902178963061, 6.55387996300723121445671205442, 7.66055136422159697744966490867, 8.391689240416621556467769802936, 9.210429988555908133950926300521

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.062634000\)
\(L(\frac12)\)	\(\approx\)	\(2.062634000\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more