LMFDB - L-function 2-528-44.43-c5-0-40

Dirichlet series

L(s) = 1

+ 9i·3^-s + 48.1·5^-s + 32.4·7^-s − 81·9^-s + (−397. + 58.0i)11^-s − 711. i·13^-s + 433. i·15^-s + 528. i·17^-s + 1.61e3·19^-s + 292. i·21^-s + 2.48e3i·23^-s − 805.·25^-s − 729i·27^-s − 3.53e3i·29^-s − 9.75e3i·31^-s + ⋯

L(s) = 1

+ 0.577i·3^-s + 0.861·5^-s + 0.250·7^-s − 0.333·9^-s + (−0.989 + 0.144i)11^-s − 1.16i·13^-s + 0.497i·15^-s + 0.443i·17^-s + 1.02·19^-s + 0.144i·21^-s + 0.979i·23^-s − 0.257·25^-s − 0.192i·27^-s − 0.781i·29^-s − 1.82i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$528$ = $2^{4} \cdot 3 \cdot 11$
Sign:	$0.929 + 0.369i$
Analytic conductor:	$84.6826$
Root analytic conductor:	$9.20231$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{528} (175, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 528,\ (\ :5/2),\ 0.929 + 0.369i)$

Particular Values

$L(3)$	$\approx$	$2.299478730$
$L(\frac12)$	$\approx$	$2.299478730$
$L(\frac{7}{2})$		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 - 9iT $
	11	$ 1 + (397. - 58.0i)T $
good	5	$ 1 - 48.1T + 3.12e3T^{2} $
	7	$ 1 - 32.4T + 1.68e4T^{2} $
	13	$ 1 + 711. iT - 3.71e5T^{2} $
	17	$ 1 - 528. iT - 1.41e6T^{2} $
	19	$ 1 - 1.61e3T + 2.47e6T^{2} $
	23	$ 1 - 2.48e3iT - 6.43e6T^{2} $
	29	$ 1 + 3.53e3iT - 2.05e7T^{2} $
	31	$ 1 + 9.75e3iT - 2.86e7T^{2} $
	37	$ 1 - 1.39e4T + 6.93e7T^{2} $
	41	$ 1 + 1.11e4iT - 1.15e8T^{2} $
	43	$ 1 - 9.93e3T + 1.47e8T^{2} $
	47	$ 1 - 1.29e4iT - 2.29e8T^{2} $
	53	$ 1 - 9.90e3T + 4.18e8T^{2} $
	59	$ 1 - 2.79e4iT - 7.14e8T^{2} $
	61	$ 1 + 5.36e4iT - 8.44e8T^{2} $
	67	$ 1 - 2.19e4iT - 1.35e9T^{2} $
	71	$ 1 + 7.36e4iT - 1.80e9T^{2} $
	73	$ 1 - 5.46e4iT - 2.07e9T^{2} $
	79	$ 1 - 6.78e4T + 3.07e9T^{2} $
	83	$ 1 + 1.04e5T + 3.93e9T^{2} $
	89	$ 1 - 5.98e4T + 5.58e9T^{2} $
	97	$ 1 + 9.83e4T + 8.58e9T^{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

−9.805771684930498547534185253577, −9.542214383232164598756424123472, −8.091053645969492266138075879541, −7.57768607343618774474849296452, −5.83339477345369347441746568134, −5.60843675359409750884336160221, −4.38192144364763458738574375470, −3.10373922736631030826452496335, −2.10346928895444896425563233158, −0.58776178587857001244820920450, 0.992240565415546104986585459498, 2.06631504334973142129816148845, 3.01304620087098429179442373029, 4.65948461511403705189602767247, 5.52166244888637800965950102799, 6.50567690034316200820189178657, 7.33149512295363490284017890264, 8.328440215410371301870249692615, 9.236659022852857021709132587058, 10.06416015483417858635131757279

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(2.299478730\)
\(L(\frac12)\)	\(\approx\)	\(2.299478730\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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