LMFDB - L-function 2-3528-21.20-c1-0-30

Dirichlet series

L(s) = 1

− 0.289·5^-s − 6.03i·11^-s + 5.46i·13^-s + 4.45·17^-s − 4.06i·19^-s − 1.29i·23^-s − 4.91·25^-s − 0.377i·29^-s − 3.57i·31^-s − 2.03·37^-s − 5.50·41^-s − 6.45·43^-s + 10.7·47^-s + 11.2i·53^-s + 1.75i·55^-s + ⋯

L(s) = 1

− 0.129·5^-s − 1.82i·11^-s + 1.51i·13^-s + 1.08·17^-s − 0.931i·19^-s − 0.269i·23^-s − 0.983·25^-s − 0.0701i·29^-s − 0.642i·31^-s − 0.333·37^-s − 0.859·41^-s − 0.984·43^-s + 1.57·47^-s + 1.55i·53^-s + 0.235i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3528$ = $2^{3} \cdot 3^{2} \cdot 7^{2}$
Sign:	$-0.239 + 0.970i$
Analytic conductor:	$28.1712$
Root analytic conductor:	$5.30765$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3528} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3528,\ (\ :1/2),\ -0.239 + 0.970i)$

Particular Values

$L(1)$	$\approx$	$1.263135665$
$L(\frac12)$	$\approx$	$1.263135665$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
	7	$ 1 $
good	5	$ 1 + 0.289T + 5T^{2} $
	11	$ 1 + 6.03iT - 11T^{2} $
	13	$ 1 - 5.46iT - 13T^{2} $
	17	$ 1 - 4.45T + 17T^{2} $
	19	$ 1 + 4.06iT - 19T^{2} $
	23	$ 1 + 1.29iT - 23T^{2} $
	29	$ 1 + 0.377iT - 29T^{2} $
	31	$ 1 + 3.57iT - 31T^{2} $
	37	$ 1 + 2.03T + 37T^{2} $
	41	$ 1 + 5.50T + 41T^{2} $
	43	$ 1 + 6.45T + 43T^{2} $
	47	$ 1 - 10.7T + 47T^{2} $
	53	$ 1 - 11.2iT - 53T^{2} $
	59	$ 1 - 1.58T + 59T^{2} $
	61	$ 1 + 11.0iT - 61T^{2} $
	67	$ 1 - 4.08T + 67T^{2} $
	71	$ 1 + 0.410iT - 71T^{2} $
	73	$ 1 + 12.8iT - 73T^{2} $
	79	$ 1 + 12.0T + 79T^{2} $
	83	$ 1 - 0.155T + 83T^{2} $
	89	$ 1 - 6.68T + 89T^{2} $
	97	$ 1 + 16.5iT - 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

−8.431322700257554923709974261598, −7.66772829920258590934522883999, −6.83180452667749412104621904560, −6.09242599597285789149822979728, −5.45643159298534028812427428838, −4.43015771983826444485679639682, −3.64454402768848164496757904032, −2.84335922901824228579838002041, −1.65069582717287031468588151503, −0.39321289682547231725498626667, 1.25677359673036505984502929700, 2.27038377019879535821181801371, 3.36805170682917968375397369324, 4.06818271598590016887723794411, 5.20279463253187003064311551868, 5.54173049934767949879090037791, 6.67073912007539257796152295818, 7.45762251780639514166904956085, 7.896671333699922406464611875656, 8.660981080214763892490465171497

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.263135665\)
\(L(\frac12)\)	\(\approx\)	\(1.263135665\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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