LMFDB - L-function 2-1800-5.4-c3-0-62

Dirichlet series

L(s) = 1

− 12i·7^-s − 64·11^-s − 58i·13^-s − 32i·17^-s + 136·19^-s − 128i·23^-s − 144·29^-s + 20·31^-s − 18i·37^-s + 288·41^-s + 200i·43^-s − 384i·47^-s + 199·49^-s + 496i·53^-s − 128·59^-s + ⋯

L(s) = 1

− 0.647i·7^-s − 1.75·11^-s − 1.23i·13^-s − 0.456i·17^-s + 1.64·19^-s − 1.16i·23^-s − 0.922·29^-s + 0.115·31^-s − 0.0799i·37^-s + 1.09·41^-s + 0.709i·43^-s − 1.19i·47^-s + 0.580·49^-s + 1.28i·53^-s − 0.282·59^-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

$L(2)$	$\approx$	$0.3932152873$
$L(\frac12)$	$\approx$	$0.3932152873$
$L(\frac{5}{2})$		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 + 12iT - 343T^{2} $
	11	$ 1 + 64T + 1.33e3T^{2} $
	13	$ 1 + 58iT - 2.19e3T^{2} $
	17	$ 1 + 32iT - 4.91e3T^{2} $
	19	$ 1 - 136T + 6.85e3T^{2} $
	23	$ 1 + 128iT - 1.21e4T^{2} $
	29	$ 1 + 144T + 2.43e4T^{2} $
	31	$ 1 - 20T + 2.97e4T^{2} $
	37	$ 1 + 18iT - 5.06e4T^{2} $
	41	$ 1 - 288T + 6.89e4T^{2} $
	43	$ 1 - 200iT - 7.95e4T^{2} $
	47	$ 1 + 384iT - 1.03e5T^{2} $
	53	$ 1 - 496iT - 1.48e5T^{2} $
	59	$ 1 + 128T + 2.05e5T^{2} $
	61	$ 1 + 458T + 2.26e5T^{2} $
	67	$ 1 + 496iT - 3.00e5T^{2} $
	71	$ 1 + 512T + 3.57e5T^{2} $
	73	$ 1 - 602iT - 3.89e5T^{2} $
	79	$ 1 + 1.10e3T + 4.93e5T^{2} $
	83	$ 1 - 704iT - 5.71e5T^{2} $
	89	$ 1 + 960T + 7.04e5T^{2} $
	97	$ 1 - 206iT - 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

−8.256462874117722207163324139719, −7.63083248840639964612880429709, −7.17840778314708130700105552394, −5.81005970875817812941579013912, −5.30332016137997398492004057829, −4.39437550231944096163133226722, −3.16130504692258571769513401708, −2.58044692364601156839772417517, −0.990242795512852736264149046519, −0.092160355158070527508660555315, 1.50098375176095871374592539682, 2.49993053539667706191837468908, 3.38778783659632234022091700824, 4.54749442780817415172929661341, 5.46279773327275600983828546674, 5.91377900059420246383667924249, 7.26721445541541534465783662721, 7.63544854211312086908901840672, 8.632662603453677885236345100363, 9.388295226244669679260101860064

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 $\F_{q}$
Belyi maps

\(L(2)\)	\(\approx\)	\(0.3932152873\)
\(L(\frac12)\)	\(\approx\)	\(0.3932152873\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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