LMFDB - L-function 2-528-44.43-c5-0-27

Dirichlet series

L(s) = 1

− 9i·3^-s + 1.95·5^-s − 158.·7^-s − 81·9^-s + (391. − 89.4i)11^-s − 135. i·13^-s − 17.5i·15^-s + 602. i·17^-s − 456.·19^-s + 1.42e3i·21^-s + 3.40e3i·23^-s − 3.12e3·25^-s + 729i·27^-s − 3.32e3i·29^-s − 3.73e3i·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.0349·5^-s − 1.22·7^-s − 0.333·9^-s + (0.974 − 0.222i)11^-s − 0.222i·13^-s − 0.0201i·15^-s + 0.505i·17^-s − 0.290·19^-s + 0.704i·21^-s + 1.34i·23^-s − 0.998·25^-s + 0.192i·27^-s − 0.733i·29^-s − 0.697i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.294i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$528$ = $2^{4} \cdot 3 \cdot 11$
Sign:	$0.955 + 0.294i$
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:	$(2,\ 528,\ (\ :5/2),\ 0.955 + 0.294i)$

Particular Values

$L(3)$	$\approx$	$1.550332171$
$L(\frac12)$	$\approx$	$1.550332171$
$L(\frac{7}{2})$		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 + (-391. + 89.4i)T $
good	5	$ 1 - 1.95T + 3.12e3T^{2} $
	7	$ 1 + 158.T + 1.68e4T^{2} $
	13	$ 1 + 135. iT - 3.71e5T^{2} $
	17	$ 1 - 602. iT - 1.41e6T^{2} $
	19	$ 1 + 456.T + 2.47e6T^{2} $
	23	$ 1 - 3.40e3iT - 6.43e6T^{2} $
	29	$ 1 + 3.32e3iT - 2.05e7T^{2} $
	31	$ 1 + 3.73e3iT - 2.86e7T^{2} $
	37	$ 1 + 9.30e3T + 6.93e7T^{2} $
	41	$ 1 - 1.39e4iT - 1.15e8T^{2} $
	43	$ 1 - 1.98e4T + 1.47e8T^{2} $
	47	$ 1 - 1.39e4iT - 2.29e8T^{2} $
	53	$ 1 - 3.41e4T + 4.18e8T^{2} $
	59	$ 1 + 2.47e4iT - 7.14e8T^{2} $
	61	$ 1 - 2.05e4iT - 8.44e8T^{2} $
	67	$ 1 + 7.11e4iT - 1.35e9T^{2} $
	71	$ 1 - 7.00e4iT - 1.80e9T^{2} $
	73	$ 1 + 3.13e4iT - 2.07e9T^{2} $
	79	$ 1 - 5.42e4T + 3.07e9T^{2} $
	83	$ 1 - 7.40e4T + 3.93e9T^{2} $
	89	$ 1 - 8.21e3T + 5.58e9T^{2} $
	97	$ 1 + 1.13e5T + 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

−9.795497787034871040991730950243, −9.287377925997718262385366530854, −8.167564478569401277316034316947, −7.23934013139801111623720801227, −6.28602149337771995802689256810, −5.76227421958774433177121340453, −4.08010579641520315543694160249, −3.22460140986291335866912768221, −1.91680676479422811494380998580, −0.65051752720180077443369527461, 0.56889863013161677574429528910, 2.25089090488200788339273395254, 3.47204989469878370933214862871, 4.23364131030817562537498385708, 5.48150105524977220714812082721, 6.50368121493307747439904017126, 7.16358664245584963273125235388, 8.699059398011345887756559248132, 9.198196183530574506845825355026, 10.09728943629669862545282712645

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(1.550332171\)
\(L(\frac12)\)	\(\approx\)	\(1.550332171\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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