LMFDB - L-function 2-1800-5.4-c3-0-48

Dirichlet series

L(s) = 1

− 5i·7^-s − 14·11^-s + i·13^-s + 46i·17^-s − 19·19^-s + 46i·23^-s + 14·29^-s + 133·31^-s − 258i·37^-s − 84·41^-s − 167i·43^-s + 410i·47^-s + 318·49^-s − 456i·53^-s − 194·59^-s + ⋯

L(s) = 1

− 0.269i·7^-s − 0.383·11^-s + 0.0213i·13^-s + 0.656i·17^-s − 0.229·19^-s + 0.417i·23^-s + 0.0896·29^-s + 0.770·31^-s − 1.14i·37^-s − 0.319·41^-s − 0.592i·43^-s + 1.27i·47^-s + 0.927·49^-s − 1.18i·53^-s − 0.428·59^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$1800$ = $2^{3} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.447 + 0.894i$
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:	$(2,\ 1800,\ (\ :3/2),\ -0.447 + 0.894i)$

Particular Values

$L(2)$	$\approx$	$1.006866187$
$L(\frac12)$	$\approx$	$1.006866187$
$L(\frac{5}{2})$		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 + 5iT - 343T^{2} $
	11	$ 1 + 14T + 1.33e3T^{2} $
	13	$ 1 - iT - 2.19e3T^{2} $
	17	$ 1 - 46iT - 4.91e3T^{2} $
	19	$ 1 + 19T + 6.85e3T^{2} $
	23	$ 1 - 46iT - 1.21e4T^{2} $
	29	$ 1 - 14T + 2.43e4T^{2} $
	31	$ 1 - 133T + 2.97e4T^{2} $
	37	$ 1 + 258iT - 5.06e4T^{2} $
	41	$ 1 + 84T + 6.89e4T^{2} $
	43	$ 1 + 167iT - 7.95e4T^{2} $
	47	$ 1 - 410iT - 1.03e5T^{2} $
	53	$ 1 + 456iT - 1.48e5T^{2} $
	59	$ 1 + 194T + 2.05e5T^{2} $
	61	$ 1 + 17T + 2.26e5T^{2} $
	67	$ 1 + 653iT - 3.00e5T^{2} $
	71	$ 1 + 828T + 3.57e5T^{2} $
	73	$ 1 - 570iT - 3.89e5T^{2} $
	79	$ 1 - 552T + 4.93e5T^{2} $
	83	$ 1 + 142iT - 5.71e5T^{2} $
	89	$ 1 + 1.10e3T + 7.04e5T^{2} $
	97	$ 1 + 841iT - 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

−8.617190674675728636225261170691, −7.88061089006616393460637474878, −7.13212757852434674979840553204, −6.23860887190096125143912679361, −5.45846272535351158481225122956, −4.48464602346167380237537251035, −3.65745635718767971466762032472, −2.59410291728196429053821993872, −1.50150977452343665718071761984, −0.23136645742809323516223790514, 1.03859485927879428294446752631, 2.34057615370752309759200190601, 3.12784551227140321830914791744, 4.32212075797690691869831929344, 5.07628384929594705147211156638, 5.97713656241324522250851016807, 6.78660211770914979109623115383, 7.62286755274128728592367266603, 8.433450826693909415053468609062, 9.082133900033455591715144742662

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.006866187\)
\(L(\frac12)\)	\(\approx\)	\(1.006866187\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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