LMFDB - L-function 2-1800-5.4-c3-0-16

Dirichlet series

L(s) = 1

− 4i·7^-s − 72·11^-s − 6i·13^-s + 38i·17^-s − 52·19^-s − 152i·23^-s − 78·29^-s + 120·31^-s + 150i·37^-s − 362·41^-s − 484i·43^-s + 280i·47^-s + 327·49^-s + 670i·53^-s + 696·59^-s + ⋯

L(s) = 1

− 0.215i·7^-s − 1.97·11^-s − 0.128i·13^-s + 0.542i·17^-s − 0.627·19^-s − 1.37i·23^-s − 0.499·29^-s + 0.695·31^-s + 0.666i·37^-s − 1.37·41^-s − 1.71i·43^-s + 0.868i·47^-s + 0.953·49^-s + 1.73i·53^-s + 1.53·59^-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

$L(2)$	$\approx$	$1.226501376$
$L(\frac12)$	$\approx$	$1.226501376$
$L(\frac{5}{2})$		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 + 4iT - 343T^{2} $
	11	$ 1 + 72T + 1.33e3T^{2} $
	13	$ 1 + 6iT - 2.19e3T^{2} $
	17	$ 1 - 38iT - 4.91e3T^{2} $
	19	$ 1 + 52T + 6.85e3T^{2} $
	23	$ 1 + 152iT - 1.21e4T^{2} $
	29	$ 1 + 78T + 2.43e4T^{2} $
	31	$ 1 - 120T + 2.97e4T^{2} $
	37	$ 1 - 150iT - 5.06e4T^{2} $
	41	$ 1 + 362T + 6.89e4T^{2} $
	43	$ 1 + 484iT - 7.95e4T^{2} $
	47	$ 1 - 280iT - 1.03e5T^{2} $
	53	$ 1 - 670iT - 1.48e5T^{2} $
	59	$ 1 - 696T + 2.05e5T^{2} $
	61	$ 1 - 222T + 2.26e5T^{2} $
	67	$ 1 - 4iT - 3.00e5T^{2} $
	71	$ 1 + 96T + 3.57e5T^{2} $
	73	$ 1 - 178iT - 3.89e5T^{2} $
	79	$ 1 - 632T + 4.93e5T^{2} $
	83	$ 1 - 612iT - 5.71e5T^{2} $
	89	$ 1 - 994T + 7.04e5T^{2} $
	97	$ 1 + 1.63e3iT - 9.12e5T^{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

−8.729506132722121740935377533976, −8.252461804679939124160849011687, −7.47366694378677567527593638023, −6.61374709425533344248739659561, −5.67865959901195474963839717366, −4.93764322837246195963958473084, −4.04455543653042023952337869989, −2.86872127922451614923331089956, −2.12657878612317860353125078668, −0.60467369287496069402479059828, 0.40935333957269288356934686551, 1.95035263958724987888613994055, 2.77872204378069124863887952772, 3.77153685162631980518425940141, 5.04527079815134221666517897767, 5.38820423823188472610113684890, 6.47945172842430259768838614561, 7.41846790654937294001570853889, 8.021432699691598130693618774215, 8.769624283951839779224161818360

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.226501376\)
\(L(\frac12)\)	\(\approx\)	\(1.226501376\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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