LMFDB - L-function 2-1344-28.27-c3-0-59

Dirichlet series

L(s) = 1

− 3·3^-s + 4.47i·5^-s + (14.9 + 10.8i)7^-s + 9·9^-s − 7.61i·11^-s + 13.3i·13^-s − 13.4i·15^-s − 55.3i·17^-s + 73.9·19^-s + (−44.9 − 32.6i)21^-s − 133. i·23^-s + 104.·25^-s − 27·27^-s − 23.3·29^-s − 241.·31^-s + ⋯

L(s) = 1

− 0.577·3^-s + 0.400i·5^-s + (0.808 + 0.587i)7^-s + 0.333·9^-s − 0.208i·11^-s + 0.285i·13^-s − 0.231i·15^-s − 0.789i·17^-s + 0.892·19^-s + (−0.467 − 0.339i)21^-s − 1.20i·23^-s + 0.839·25^-s − 0.192·27^-s − 0.149·29^-s − 1.39·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.587i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.808 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$0.808 + 0.587i$
Analytic conductor:	$79.2985$
Root analytic conductor:	$8.90497$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1344} (895, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1344,\ (\ :3/2),\ 0.808 + 0.587i)$

Particular Values

$L(2)$	$\approx$	$1.719334107$
$L(\frac12)$	$\approx$	$1.719334107$
$L(\frac{5}{2})$		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 $
	3	$ 1 + 3T $
	7	$ 1 + (-14.9 - 10.8i)T $
good	5	$ 1 - 4.47iT - 125T^{2} $
	11	$ 1 + 7.61iT - 1.33e3T^{2} $
	13	$ 1 - 13.3iT - 2.19e3T^{2} $
	17	$ 1 + 55.3iT - 4.91e3T^{2} $
	19	$ 1 - 73.9T + 6.85e3T^{2} $
	23	$ 1 + 133. iT - 1.21e4T^{2} $
	29	$ 1 + 23.3T + 2.43e4T^{2} $
	31	$ 1 + 241.T + 2.97e4T^{2} $
	37	$ 1 + 178.T + 5.06e4T^{2} $
	41	$ 1 + 494. iT - 6.89e4T^{2} $
	43	$ 1 - 72.6iT - 7.95e4T^{2} $
	47	$ 1 - 59.7T + 1.03e5T^{2} $
	53	$ 1 - 569.T + 1.48e5T^{2} $
	59	$ 1 + 59.5T + 2.05e5T^{2} $
	61	$ 1 + 629. iT - 2.26e5T^{2} $
	67	$ 1 + 599. iT - 3.00e5T^{2} $
	71	$ 1 - 407. iT - 3.57e5T^{2} $
	73	$ 1 - 680. iT - 3.89e5T^{2} $
	79	$ 1 - 1.08e3iT - 4.93e5T^{2} $
	83	$ 1 + 935.T + 5.71e5T^{2} $
	89	$ 1 - 12.1iT - 7.04e5T^{2} $
	97	$ 1 + 1.43e3iT - 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

−9.087846684215149212967884759698, −8.454022438034771464726549106652, −7.32078879568033157782400703624, −6.83124675344713750468765444753, −5.62268589008880664874581299322, −5.16610292831680297265339537127, −4.11008824898193398610747184788, −2.88870782909482381212992105503, −1.83852687413677039519228246482, −0.52145347446160714668521918472, 0.953987175299211555789793835914, 1.73653638320281169260902803495, 3.36313047743202579362540308292, 4.32727364344263285162119361841, 5.18453642296795208179271202501, 5.79154892186140663566839571157, 7.03726935347128281208415276472, 7.58230368084627401548569711732, 8.475333467577384681864945531606, 9.349989225691462026937067511256

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.719334107\)
\(L(\frac12)\)	\(\approx\)	\(1.719334107\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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