LMFDB - L-function 2-288-24.11-c7-0-15

Dirichlet series

L(s) = 1

− 527.·5^-s − 1.31e3i·7^-s + 961. i·11^-s + 1.07e4i·13^-s + 1.15e4i·17^-s + 4.69e4·19^-s + 8.87e3·23^-s + 2.00e5·25^-s − 1.09e5·29^-s + 2.39e5i·31^-s + 6.92e5i·35^-s − 3.66e5i·37^-s − 4.07e4i·41^-s − 2.25e5·43^-s − 7.13e5·47^-s + ⋯

L(s) = 1

− 1.88·5^-s − 1.44i·7^-s + 0.217i·11^-s + 1.35i·13^-s + 0.569i·17^-s + 1.56·19^-s + 0.152·23^-s + 2.56·25^-s − 0.833·29^-s + 1.44i·31^-s + 2.73i·35^-s − 1.18i·37^-s − 0.0922i·41^-s − 0.432·43^-s − 1.00·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(8-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$288$ = $2^{5} \cdot 3^{2}$
Sign:	$-0.298 + 0.954i$
Analytic conductor:	$89.9668$
Root analytic conductor:	$9.48508$
Motivic weight:	$7$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{288} (143, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 288,\ (\ :7/2),\ -0.298 + 0.954i)$

Particular Values

$L(4)$	$\approx$	$0.7436154523$
$L(\frac12)$	$\approx$	$0.7436154523$
$L(\frac{9}{2})$		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 + 527.T + 7.81e4T^{2} $
	7	$ 1 + 1.31e3iT - 8.23e5T^{2} $
	11	$ 1 - 961. iT - 1.94e7T^{2} $
	13	$ 1 - 1.07e4iT - 6.27e7T^{2} $
	17	$ 1 - 1.15e4iT - 4.10e8T^{2} $
	19	$ 1 - 4.69e4T + 8.93e8T^{2} $
	23	$ 1 - 8.87e3T + 3.40e9T^{2} $
	29	$ 1 + 1.09e5T + 1.72e10T^{2} $
	31	$ 1 - 2.39e5iT - 2.75e10T^{2} $
	37	$ 1 + 3.66e5iT - 9.49e10T^{2} $
	41	$ 1 + 4.07e4iT - 1.94e11T^{2} $
	43	$ 1 + 2.25e5T + 2.71e11T^{2} $
	47	$ 1 + 7.13e5T + 5.06e11T^{2} $
	53	$ 1 - 7.86e5T + 1.17e12T^{2} $
	59	$ 1 + 1.19e6iT - 2.48e12T^{2} $
	61	$ 1 + 1.03e6iT - 3.14e12T^{2} $
	67	$ 1 - 1.72e6T + 6.06e12T^{2} $
	71	$ 1 - 8.27e4T + 9.09e12T^{2} $
	73	$ 1 + 2.92e6T + 1.10e13T^{2} $
	79	$ 1 + 1.21e6iT - 1.92e13T^{2} $
	83	$ 1 - 1.39e6iT - 2.71e13T^{2} $
	89	$ 1 - 1.64e6iT - 4.42e13T^{2} $
	97	$ 1 - 1.75e5T + 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

−10.53658131634232765335079817198, −9.306343616730230970875210820760, −8.167344788694806633399618526437, −7.30584309240276866989549945988, −6.88228903715810003012978584522, −4.91366350880800857005662447043, −3.98998684726518326397806402514, −3.43323272298842222398364565097, −1.36250413185705445335253372247, −0.24749655279037002246555046596, 0.815016382931148124507660544489, 2.78427829205274067511112738213, 3.48557565355160572249155063707, 4.88361924055566948523393120611, 5.76106832592503264226635011130, 7.29249947835301571336852248092, 7.993769850621114592517563395891, 8.742171511837809415130855079923, 9.861616448089567810088957694144, 11.30162108860865757641467056567

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 $\F_{q}$

\(L(4)\)	\(\approx\)	\(0.7436154523\)
\(L(\frac12)\)	\(\approx\)	\(0.7436154523\)
\(L(\frac{9}{2})\)		not available
\(L(1)\)		not available

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