LMFDB - L-function 2-2800-5.4-c1-0-21

Dirichlet series

L(s) = 1

− 3i·3^-s + i·7^-s − 6·9^-s + 5·11^-s + 6i·13^-s − i·17^-s − 3·19^-s + 3·21^-s + 9i·27^-s + 6·29^-s + 4·31^-s − 15i·33^-s + 8i·37^-s + 18·39^-s + 11·41^-s + ⋯

L(s) = 1

− 1.73i·3^-s + 0.377i·7^-s − 2·9^-s + 1.50·11^-s + 1.66i·13^-s − 0.242i·17^-s − 0.688·19^-s + 0.654·21^-s + 1.73i·27^-s + 1.11·29^-s + 0.718·31^-s − 2.61i·33^-s + 1.31i·37^-s + 2.88·39^-s + 1.71·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2800$ = $2^{4} \cdot 5^{2} \cdot 7$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$22.3581$
Root analytic conductor:	$4.72843$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2800} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2800,\ (\ :1/2),\ 0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$1.919762498$
$L(\frac12)$	$\approx$	$1.919762498$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
	7	$ 1 - iT $
good	3	$ 1 + 3iT - 3T^{2} $
	11	$ 1 - 5T + 11T^{2} $
	13	$ 1 - 6iT - 13T^{2} $
	17	$ 1 + iT - 17T^{2} $
	19	$ 1 + 3T + 19T^{2} $
	23	$ 1 - 23T^{2} $
	29	$ 1 - 6T + 29T^{2} $
	31	$ 1 - 4T + 31T^{2} $
	37	$ 1 - 8iT - 37T^{2} $
	41	$ 1 - 11T + 41T^{2} $
	43	$ 1 + 8iT - 43T^{2} $
	47	$ 1 + 2iT - 47T^{2} $
	53	$ 1 + 4iT - 53T^{2} $
	59	$ 1 - 4T + 59T^{2} $
	61	$ 1 + 2T + 61T^{2} $
	67	$ 1 + 9iT - 67T^{2} $
	71	$ 1 - 10T + 71T^{2} $
	73	$ 1 - 7iT - 73T^{2} $
	79	$ 1 + 2T + 79T^{2} $
	83	$ 1 - 11iT - 83T^{2} $
	89	$ 1 - 11T + 89T^{2} $
	97	$ 1 + 10iT - 97T^{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

−8.680502964898596526145521334914, −7.85526139618735009564756144543, −6.85712782025260333711457713729, −6.60881873513603227130121832174, −6.04844473627330597333054081040, −4.77994703781332761587988288891, −3.83965706151057570278237292129, −2.54685890465382673269306460434, −1.81937509470877097482148290782, −0.930520821249209750120031638115, 0.849514495136806981317882134586, 2.66749320425673017600255589050, 3.53629591371083024295916370386, 4.19450755223027610189939265590, 4.80025853461378712326735180606, 5.81915723962884947796499165911, 6.35568638090845137880446584403, 7.60162871113490771013261133864, 8.416205533675640242505669875373, 9.076235760795153340601652790456

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.919762498\)
\(L(\frac12)\)	\(\approx\)	\(1.919762498\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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