LMFDB - L-function 2-528-44.43-c5-0-4

Dirichlet series

L(s) = 1

− 9i·3^-s + 89.8·5^-s − 112.·7^-s − 81·9^-s + (−346. − 202. i)11^-s − 898. i·13^-s − 808. i·15^-s − 13.3i·17^-s − 2.20e3·19^-s + 1.00e3i·21^-s + 2.28e3i·23^-s + 4.94e3·25^-s + 729i·27^-s + 2.52e3i·29^-s + 6.30e3i·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 1.60·5^-s − 0.864·7^-s − 0.333·9^-s + (−0.863 − 0.503i)11^-s − 1.47i·13^-s − 0.927i·15^-s − 0.0112i·17^-s − 1.40·19^-s + 0.499i·21^-s + 0.901i·23^-s + 1.58·25^-s + 0.192i·27^-s + 0.558i·29^-s + 1.17i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.496 - 0.868i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.496 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$528$ = $2^{4} \cdot 3 \cdot 11$
Sign:	$-0.496 - 0.868i$
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:	$(2,\ 528,\ (\ :5/2),\ -0.496 - 0.868i)$

Particular Values

$L(3)$	$\approx$	$0.2471788845$
$L(\frac12)$	$\approx$	$0.2471788845$
$L(\frac{7}{2})$		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 + (346. + 202. i)T $
good	5	$ 1 - 89.8T + 3.12e3T^{2} $
	7	$ 1 + 112.T + 1.68e4T^{2} $
	13	$ 1 + 898. iT - 3.71e5T^{2} $
	17	$ 1 + 13.3iT - 1.41e6T^{2} $
	19	$ 1 + 2.20e3T + 2.47e6T^{2} $
	23	$ 1 - 2.28e3iT - 6.43e6T^{2} $
	29	$ 1 - 2.52e3iT - 2.05e7T^{2} $
	31	$ 1 - 6.30e3iT - 2.86e7T^{2} $
	37	$ 1 - 6.83e3T + 6.93e7T^{2} $
	41	$ 1 + 7.88e3iT - 1.15e8T^{2} $
	43	$ 1 - 77.7T + 1.47e8T^{2} $
	47	$ 1 - 1.77e4iT - 2.29e8T^{2} $
	53	$ 1 + 3.34e4T + 4.18e8T^{2} $
	59	$ 1 - 6.09e3iT - 7.14e8T^{2} $
	61	$ 1 - 4.02e4iT - 8.44e8T^{2} $
	67	$ 1 - 3.07e4iT - 1.35e9T^{2} $
	71	$ 1 + 1.33e4iT - 1.80e9T^{2} $
	73	$ 1 + 5.45e4iT - 2.07e9T^{2} $
	79	$ 1 - 1.04e5T + 3.07e9T^{2} $
	83	$ 1 - 1.22e4T + 3.93e9T^{2} $
	89	$ 1 + 1.05e5T + 5.58e9T^{2} $
	97	$ 1 - 2.32e4T + 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

−10.39353986669960883905371787096, −9.531645426262306613259416314461, −8.637745488040150483134322495289, −7.64616306867927102480203117035, −6.45817113298689899653949005452, −5.89381161237133889850490594007, −5.12619150552302589337139261393, −3.20590273522914266127846336997, −2.47454671458004576550724465291, −1.25200387626488490973188095884, 0.05063162525794051235456991849, 1.94821454896355856201974165012, 2.60120776434863162970863593177, 4.14661781371887820871789963141, 5.05896022229902138591070529825, 6.27763807364461348341654836816, 6.55487160986060115132211083354, 8.161209231784274449290563041190, 9.325809210532033800896422538094, 9.656217138906988763764886325252

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(0.2471788845\)
\(L(\frac12)\)	\(\approx\)	\(0.2471788845\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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