LMFDB - L-function 2-1800-5.4-c3-0-29

Dirichlet series

L(s) = 1

− 18i·7^-s − 34·11^-s − 12i·13^-s + 102i·17^-s − 164·19^-s + 48i·23^-s + 146·29^-s + 100·31^-s + 328i·37^-s + 288·41^-s − 120i·43^-s − 16i·47^-s + 19·49^-s − 126i·53^-s + 642·59^-s + ⋯

L(s) = 1

− 0.971i·7^-s − 0.931·11^-s − 0.256i·13^-s + 1.45i·17^-s − 1.98·19^-s + 0.435i·23^-s + 0.934·29^-s + 0.579·31^-s + 1.45i·37^-s + 1.09·41^-s − 0.425i·43^-s − 0.0496i·47^-s + 0.0553·49^-s − 0.326i·53^-s + 1.41·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1800$ = $2^{3} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$106.203$
Root analytic conductor:	$10.3055$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1800} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1800,\ (\ :3/2),\ 0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$1.566362853$
$L(\frac12)$	$\approx$	$1.566362853$
$L(\frac{5}{2})$		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 $
	5	$ 1 $
good	7	$ 1 + 18iT - 343T^{2} $
	11	$ 1 + 34T + 1.33e3T^{2} $
	13	$ 1 + 12iT - 2.19e3T^{2} $
	17	$ 1 - 102iT - 4.91e3T^{2} $
	19	$ 1 + 164T + 6.85e3T^{2} $
	23	$ 1 - 48iT - 1.21e4T^{2} $
	29	$ 1 - 146T + 2.43e4T^{2} $
	31	$ 1 - 100T + 2.97e4T^{2} $
	37	$ 1 - 328iT - 5.06e4T^{2} $
	41	$ 1 - 288T + 6.89e4T^{2} $
	43	$ 1 + 120iT - 7.95e4T^{2} $
	47	$ 1 + 16iT - 1.03e5T^{2} $
	53	$ 1 + 126iT - 1.48e5T^{2} $
	59	$ 1 - 642T + 2.05e5T^{2} $
	61	$ 1 - 602T + 2.26e5T^{2} $
	67	$ 1 - 436iT - 3.00e5T^{2} $
	71	$ 1 + 652T + 3.57e5T^{2} $
	73	$ 1 + 1.06e3iT - 3.89e5T^{2} $
	79	$ 1 + 388T + 4.93e5T^{2} $
	83	$ 1 + 444iT - 5.71e5T^{2} $
	89	$ 1 + 820T + 7.04e5T^{2} $
	97	$ 1 + 766iT - 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

−8.474570176457689265467335617286, −8.298440517551279184321806326385, −7.27472571411688537391920286952, −6.48287861029054003941527227230, −5.72066868292288754800797770956, −4.56579016729914422942613070712, −3.99792749438147756218544294994, −2.86089070186303507763398681482, −1.76869132634985045319625941025, −0.52895215832126077603487269289, 0.62031276665215119000420308960, 2.35461074134126872567380362613, 2.60794500004656149147976472899, 4.10953731703821525179483119604, 4.92880324489572581948573605293, 5.73346532564049712022260755258, 6.54632833699480558596868875106, 7.39823030115990895305852037777, 8.374249994271532561645647958548, 8.832462985108233802618440123180

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.566362853\)
\(L(\frac12)\)	\(\approx\)	\(1.566362853\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more