LMFDB - L-function 2-448-8.5-c3-0-4

Dirichlet series

L(s) = 1

− 9.66i·3^-s + 11.8i·5^-s − 7·7^-s − 66.3·9^-s + 36.1i·11^-s − 59.5i·13^-s + 114.·15^-s − 28.4·17^-s + 128. i·19^-s + 67.6i·21^-s + 168.·23^-s − 14.3·25^-s + 379. i·27^-s + 24.8i·29^-s + 98.4·31^-s + ⋯

L(s) = 1

− 1.85i·3^-s + 1.05i·5^-s − 0.377·7^-s − 2.45·9^-s + 0.989i·11^-s − 1.26i·13^-s + 1.96·15^-s − 0.405·17^-s + 1.54i·19^-s + 0.702i·21^-s + 1.52·23^-s − 0.114·25^-s + 2.70i·27^-s + 0.159i·29^-s + 0.570·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$448$ = $2^{6} \cdot 7$
Sign:	$0.965 - 0.258i$
Analytic conductor:	$26.4328$
Root analytic conductor:	$5.14128$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{448} (225, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 448,\ (\ :3/2),\ 0.965 - 0.258i)$

Particular Values

$L(2)$	$\approx$	$1.310021325$
$L(\frac12)$	$\approx$	$1.310021325$
$L(\frac{5}{2})$		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 $
	7	$ 1 + 7T $
good	3	$ 1 + 9.66iT - 27T^{2} $
	5	$ 1 - 11.8iT - 125T^{2} $
	11	$ 1 - 36.1iT - 1.33e3T^{2} $
	13	$ 1 + 59.5iT - 2.19e3T^{2} $
	17	$ 1 + 28.4T + 4.91e3T^{2} $
	19	$ 1 - 128. iT - 6.85e3T^{2} $
	23	$ 1 - 168.T + 1.21e4T^{2} $
	29	$ 1 - 24.8iT - 2.43e4T^{2} $
	31	$ 1 - 98.4T + 2.97e4T^{2} $
	37	$ 1 - 411. iT - 5.06e4T^{2} $
	41	$ 1 + 302.T + 6.89e4T^{2} $
	43	$ 1 + 61.8iT - 7.95e4T^{2} $
	47	$ 1 - 493.T + 1.03e5T^{2} $
	53	$ 1 + 66.6iT - 1.48e5T^{2} $
	59	$ 1 + 467. iT - 2.05e5T^{2} $
	61	$ 1 - 315. iT - 2.26e5T^{2} $
	67	$ 1 - 765. iT - 3.00e5T^{2} $
	71	$ 1 - 241.T + 3.57e5T^{2} $
	73	$ 1 - 413.T + 3.89e5T^{2} $
	79	$ 1 - 711.T + 4.93e5T^{2} $
	83	$ 1 + 147. iT - 5.71e5T^{2} $
	89	$ 1 + 425.T + 7.04e5T^{2} $
	97	$ 1 + 1.13e3T + 9.12e5T^{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

−10.79462735586476107754906849814, −9.998967726309375730302891819541, −8.542251614489064108934395022037, −7.72957646232937544055915110895, −6.95570892124293237947612448222, −6.43445186697888168894355703773, −5.33888251068189840874242508825, −3.26882376490822634159865755793, −2.44200337852162393570086843706, −1.14253313550994790905638192966, 0.47808367271363901489468562474, 2.79549638213313871063582555833, 3.97941457016640019111177580940, 4.74793187066454165833480389282, 5.48161204123068506809442082323, 6.74859142880018561343366325730, 8.505579671419432453734860205829, 9.179505070820597702729529166442, 9.332233631117183444233425624763, 10.79634026873518485732861044020

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.310021325\)
\(L(\frac12)\)	\(\approx\)	\(1.310021325\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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