LMFDB - L-function 2-1344-8.5-c3-0-27

Dirichlet series

L(s) = 1

− 3i·3^-s + 7.03i·5^-s + 7·7^-s − 9·9^-s + 46.7i·11^-s + 8.90i·13^-s + 21.1·15^-s + 58.0·17^-s − 135. i·19^-s − 21i·21^-s − 86.0·23^-s + 75.5·25^-s + 27i·27^-s − 300. i·29^-s + 183.·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.629i·5^-s + 0.377·7^-s − 0.333·9^-s + 1.28i·11^-s + 0.190i·13^-s + 0.363·15^-s + 0.828·17^-s − 1.63i·19^-s − 0.218i·21^-s − 0.779·23^-s + 0.604·25^-s + 0.192i·27^-s − 1.92i·29^-s + 1.06·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$79.2985$
Root analytic conductor:	$8.90497$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1344} (673, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1344,\ (\ :3/2),\ 0.707 - 0.707i)$

Particular Values

$L(2)$	$\approx$	$2.040248665$
$L(\frac12)$	$\approx$	$2.040248665$
$L(\frac{5}{2})$		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 + 3iT $
	7	$ 1 - 7T $
good	5	$ 1 - 7.03iT - 125T^{2} $
	11	$ 1 - 46.7iT - 1.33e3T^{2} $
	13	$ 1 - 8.90iT - 2.19e3T^{2} $
	17	$ 1 - 58.0T + 4.91e3T^{2} $
	19	$ 1 + 135. iT - 6.85e3T^{2} $
	23	$ 1 + 86.0T + 1.21e4T^{2} $
	29	$ 1 + 300. iT - 2.43e4T^{2} $
	31	$ 1 - 183.T + 2.97e4T^{2} $
	37	$ 1 - 418. iT - 5.06e4T^{2} $
	41	$ 1 + 106.T + 6.89e4T^{2} $
	43	$ 1 - 162. iT - 7.95e4T^{2} $
	47	$ 1 + 377.T + 1.03e5T^{2} $
	53	$ 1 - 709. iT - 1.48e5T^{2} $
	59	$ 1 + 753. iT - 2.05e5T^{2} $
	61	$ 1 - 667. iT - 2.26e5T^{2} $
	67	$ 1 - 447. iT - 3.00e5T^{2} $
	71	$ 1 - 384.T + 3.57e5T^{2} $
	73	$ 1 - 1.09e3T + 3.89e5T^{2} $
	79	$ 1 - 670.T + 4.93e5T^{2} $
	83	$ 1 - 447. iT - 5.71e5T^{2} $
	89	$ 1 + 70.5T + 7.04e5T^{2} $
	97	$ 1 + 1.31e3T + 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

−9.518269321189926587261461939799, −8.274018599951519536482148598612, −7.73105884205369371957199257439, −6.81677466365344123996117202245, −6.35409649953521726390760437378, −5.06032822084419354902002345855, −4.33982936134228476261603737008, −2.94168915087885847971282409645, −2.18870158243757033794223627887, −0.973843459359573591020063951289, 0.55952634024298327087978928516, 1.69394590256010139669944487414, 3.24289486135001136117661048540, 3.85276165867539497568737098049, 5.10222059699987117312791812933, 5.55218773027852160407926704987, 6.51730020663246692666405863891, 7.86849975458513666879477499109, 8.346578207925426975205758533450, 9.049080473831601354791815056388

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.040248665\)
\(L(\frac12)\)	\(\approx\)	\(2.040248665\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more