LMFDB - L-function 2-528-44.43-c5-0-18

Dirichlet series

L(s) = 1

− 9i·3^-s + 57.6·5^-s + 131.·7^-s − 81·9^-s + (−230. + 328. i)11^-s − 558. i·13^-s − 519. i·15^-s + 2.07e3i·17^-s − 347.·19^-s − 1.18e3i·21^-s + 3.83e3i·23^-s + 203.·25^-s + 729i·27^-s + 4.84e3i·29^-s + 852. i·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 1.03·5^-s + 1.01·7^-s − 0.333·9^-s + (−0.574 + 0.818i)11^-s − 0.916i·13^-s − 0.595i·15^-s + 1.74i·17^-s − 0.220·19^-s − 0.584i·21^-s + 1.51i·23^-s + 0.0649·25^-s + 0.192i·27^-s + 1.06i·29^-s + 0.159i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0884 - 0.996i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0884 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$528$ = $2^{4} \cdot 3 \cdot 11$
Sign:	$0.0884 - 0.996i$
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:	$(2,\ 528,\ (\ :5/2),\ 0.0884 - 0.996i)$

Particular Values

$L(3)$	$\approx$	$1.898345958$
$L(\frac12)$	$\approx$	$1.898345958$
$L(\frac{7}{2})$		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 + (230. - 328. i)T $
good	5	$ 1 - 57.6T + 3.12e3T^{2} $
	7	$ 1 - 131.T + 1.68e4T^{2} $
	13	$ 1 + 558. iT - 3.71e5T^{2} $
	17	$ 1 - 2.07e3iT - 1.41e6T^{2} $
	19	$ 1 + 347.T + 2.47e6T^{2} $
	23	$ 1 - 3.83e3iT - 6.43e6T^{2} $
	29	$ 1 - 4.84e3iT - 2.05e7T^{2} $
	31	$ 1 - 852. iT - 2.86e7T^{2} $
	37	$ 1 + 1.46e4T + 6.93e7T^{2} $
	41	$ 1 - 1.49e4iT - 1.15e8T^{2} $
	43	$ 1 - 2.14e3T + 1.47e8T^{2} $
	47	$ 1 + 8.29e3iT - 2.29e8T^{2} $
	53	$ 1 + 3.37e4T + 4.18e8T^{2} $
	59	$ 1 - 741. iT - 7.14e8T^{2} $
	61	$ 1 + 4.81e4iT - 8.44e8T^{2} $
	67	$ 1 - 1.81e4iT - 1.35e9T^{2} $
	71	$ 1 + 4.40e4iT - 1.80e9T^{2} $
	73	$ 1 - 4.53e3iT - 2.07e9T^{2} $
	79	$ 1 + 4.47e4T + 3.07e9T^{2} $
	83	$ 1 - 8.16e4T + 3.93e9T^{2} $
	89	$ 1 + 2.82e4T + 5.58e9T^{2} $
	97	$ 1 - 4.34e4T + 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

−10.37520842550698961858675778417, −9.430322984684050238384402372415, −8.265720144077184521705267548576, −7.74141088936934570924403091317, −6.58175113398259167512993673919, −5.59788629106497964448939771535, −4.94330442618698373280677587702, −3.36407913053575122995664365814, −1.88577020724422243120537136346, −1.52045531638639394477788486105, 0.38053701499999622733633277888, 1.89542569989704068018005766695, 2.80802358300800317865907536252, 4.34889935112038375653696812783, 5.13138978661696145516012056057, 5.97158580482837261704565362546, 7.09104703233242454093745738770, 8.290788662845268631382931562790, 9.027832227317811127971783023973, 9.846161420990845245737631943341

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(1.898345958\)
\(L(\frac12)\)	\(\approx\)	\(1.898345958\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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