LMFDB - L-function 2-288-8.5-c7-0-28

Dirichlet series

L(s) = 1

+ 23.0i·5^-s + 1.54e3·7^-s − 822. i·11^-s − 5.92e3i·13^-s + 1.27e4·17^-s − 4.38e4i·19^-s − 7.76e4·23^-s + 7.75e4·25^-s − 1.51e5i·29^-s + 4.64e4·31^-s + 3.56e4i·35^-s + 3.24e4i·37^-s − 1.88e5·41^-s + 8.47e5i·43^-s − 1.23e6·47^-s + ⋯

L(s) = 1

+ 0.0823i·5^-s + 1.70·7^-s − 0.186i·11^-s − 0.748i·13^-s + 0.628·17^-s − 1.46i·19^-s − 1.33·23^-s + 0.993·25^-s − 1.15i·29^-s + 0.279·31^-s + 0.140i·35^-s + 0.105i·37^-s − 0.427·41^-s + 1.62i·43^-s − 1.73·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(8-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$288$ = $2^{5} \cdot 3^{2}$
Sign:	$0.270 + 0.962i$
Analytic conductor:	$89.9668$
Root analytic conductor:	$9.48508$
Motivic weight:	$7$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{288} (145, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 288,\ (\ :7/2),\ 0.270 + 0.962i)$

Particular Values

$L(4)$	$\approx$	$2.471332120$
$L(\frac12)$	$\approx$	$2.471332120$
$L(\frac{9}{2})$		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 $
good	5	$ 1 - 23.0iT - 7.81e4T^{2} $
	7	$ 1 - 1.54e3T + 8.23e5T^{2} $
	11	$ 1 + 822. iT - 1.94e7T^{2} $
	13	$ 1 + 5.92e3iT - 6.27e7T^{2} $
	17	$ 1 - 1.27e4T + 4.10e8T^{2} $
	19	$ 1 + 4.38e4iT - 8.93e8T^{2} $
	23	$ 1 + 7.76e4T + 3.40e9T^{2} $
	29	$ 1 + 1.51e5iT - 1.72e10T^{2} $
	31	$ 1 - 4.64e4T + 2.75e10T^{2} $
	37	$ 1 - 3.24e4iT - 9.49e10T^{2} $
	41	$ 1 + 1.88e5T + 1.94e11T^{2} $
	43	$ 1 - 8.47e5iT - 2.71e11T^{2} $
	47	$ 1 + 1.23e6T + 5.06e11T^{2} $
	53	$ 1 - 1.00e6iT - 1.17e12T^{2} $
	59	$ 1 + 2.08e6iT - 2.48e12T^{2} $
	61	$ 1 + 2.52e5iT - 3.14e12T^{2} $
	67	$ 1 + 4.04e6iT - 6.06e12T^{2} $
	71	$ 1 + 2.06e6T + 9.09e12T^{2} $
	73	$ 1 - 1.18e6T + 1.10e13T^{2} $
	79	$ 1 - 3.44e6T + 1.92e13T^{2} $
	83	$ 1 + 4.07e6iT - 2.71e13T^{2} $
	89	$ 1 - 1.21e7T + 4.42e13T^{2} $
	97	$ 1 + 1.02e7T + 8.07e13T^{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

−10.56849072190219123388129595437, −9.440593553787233413959374524988, −8.172555380189834485581320151373, −7.83818078067441495541426305024, −6.42886887873683143537539800527, −5.20806799082047912141305790325, −4.48158409868546244750155280657, −2.96961928810673805558435530285, −1.72894636162073952480425605168, −0.56577555380209492170961536785, 1.26763457817694829784538027804, 2.01248071527803927422619132110, 3.72940970024101323794487523732, 4.76956721389111979149094777304, 5.65667065373464553063678580439, 7.01999899758824355134088158346, 8.043263179171889353500206072582, 8.639549931220777628136095848620, 9.965238805060928854521432181552, 10.78230839375468749999091364788

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(2.471332120\)
\(L(\frac12)\)	\(\approx\)	\(2.471332120\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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