LMFDB - L-function 2-4056-13.12-c1-0-4

Dirichlet series

L(s) = 1

− 3^-s + 3.80i·5^-s − 5.13i·7^-s + 9^-s − 0.334i·11^-s − 3.80i·15^-s − 4.13·17^-s − 5.94i·19^-s + 5.13i·21^-s − 0.334·23^-s − 9.47·25^-s − 27^-s − 0.195·29^-s + 4.80i·31^-s + 0.334i·33^-s + ⋯

L(s) = 1

− 0.577·3^-s + 1.70i·5^-s − 1.94i·7^-s + 0.333·9^-s − 0.100i·11^-s − 0.982i·15^-s − 1.00·17^-s − 1.36i·19^-s + 1.12i·21^-s − 0.0698·23^-s − 1.89·25^-s − 0.192·27^-s − 0.0363·29^-s + 0.862i·31^-s + 0.0582i·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4056$ = $2^{3} \cdot 3 \cdot 13^{2}$
Sign:	$-0.554 - 0.832i$
Analytic conductor:	$32.3873$
Root analytic conductor:	$5.69098$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4056} (337, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4056,\ (\ :1/2),\ -0.554 - 0.832i)$

Particular Values

$L(1)$	$\approx$	$0.6461162858$
$L(\frac12)$	$\approx$	$0.6461162858$
$L(\frac{3}{2})$		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 + T $
	13	$ 1 $
good	5	$ 1 - 3.80iT - 5T^{2} $
	7	$ 1 + 5.13iT - 7T^{2} $
	11	$ 1 + 0.334iT - 11T^{2} $
	17	$ 1 + 4.13T + 17T^{2} $
	19	$ 1 + 5.94iT - 19T^{2} $
	23	$ 1 + 0.334T + 23T^{2} $
	29	$ 1 + 0.195T + 29T^{2} $
	31	$ 1 - 4.80iT - 31T^{2} $
	37	$ 1 - 2.13iT - 37T^{2} $
	41	$ 1 - 3.46iT - 41T^{2} $
	43	$ 1 - 2.86T + 43T^{2} $
	47	$ 1 - 3.66iT - 47T^{2} $
	53	$ 1 - 9.41T + 53T^{2} $
	59	$ 1 - 6.27iT - 59T^{2} $
	61	$ 1 + 6.94T + 61T^{2} $
	67	$ 1 - 13.1iT - 67T^{2} $
	71	$ 1 - 4.33iT - 71T^{2} $
	73	$ 1 - 10.6iT - 73T^{2} $
	79	$ 1 + 0.134T + 79T^{2} $
	83	$ 1 + 13.2iT - 83T^{2} $
	89	$ 1 + 0.390iT - 89T^{2} $
	97	$ 1 - 10.8iT - 97T^{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

−8.630744693885063423280506795545, −7.49859777647556059555662176032, −7.11599426696359438989744041845, −6.71920048847174809509943402126, −6.01294553001309350091571511308, −4.75374123472213745046646353458, −4.14192347461549325542775835977, −3.31394948679637638342325092007, −2.45415788013483481117057706988, −1.03782175870250077972096546024, 0.22370731480916464827012271222, 1.65962899894441374258369332099, 2.28328209026800020561620087986, 3.74898868184369707262172114265, 4.65961774506325335101078598254, 5.25645134628765644420866594704, 5.83369812010616674972123596822, 6.35176102961494063774794786424, 7.69962796350154028671529411194, 8.326124397606718031197466899324

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.6461162858\)
\(L(\frac12)\)	\(\approx\)	\(0.6461162858\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more