LMFDB - L-function 2-1344-56.27-c3-0-45

Dirichlet series

L(s) = 1

− 3i·3^-s − 15.4·5^-s + (−6.29 + 17.4i)7^-s − 9·9^-s − 16.8·11^-s − 9.10·13^-s + 46.2i·15^-s − 73.9i·17^-s + 151. i·19^-s + (52.2 + 18.8i)21^-s − 0.264i·23^-s + 112.·25^-s + 27i·27^-s + 279. i·29^-s − 147.·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 1.37·5^-s + (−0.339 + 0.940i)7^-s − 0.333·9^-s − 0.462·11^-s − 0.194·13^-s + 0.795i·15^-s − 1.05i·17^-s + 1.82i·19^-s + (0.543 + 0.196i)21^-s − 0.00239i·23^-s + 0.897·25^-s + 0.192i·27^-s + 1.79i·29^-s − 0.856·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0847 + 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0847 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1344$ = $2^{6} \cdot 3 \cdot 7$
Sign:	$-0.0847 + 0.996i$
Analytic conductor:	$79.2985$
Root analytic conductor:	$8.90497$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1344} (223, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1344,\ (\ :3/2),\ -0.0847 + 0.996i)$

Particular Values

$L(2)$	$\approx$	$0.3682207132$
$L(\frac12)$	$\approx$	$0.3682207132$
$L(\frac{5}{2})$		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 + 3iT $
	7	$ 1 + (6.29 - 17.4i)T $
good	5	$ 1 + 15.4T + 125T^{2} $
	11	$ 1 + 16.8T + 1.33e3T^{2} $
	13	$ 1 + 9.10T + 2.19e3T^{2} $
	17	$ 1 + 73.9iT - 4.91e3T^{2} $
	19	$ 1 - 151. iT - 6.85e3T^{2} $
	23	$ 1 + 0.264iT - 1.21e4T^{2} $
	29	$ 1 - 279. iT - 2.43e4T^{2} $
	31	$ 1 + 147.T + 2.97e4T^{2} $
	37	$ 1 - 418. iT - 5.06e4T^{2} $
	41	$ 1 + 164. iT - 6.89e4T^{2} $
	43	$ 1 + 266.T + 7.95e4T^{2} $
	47	$ 1 - 277.T + 1.03e5T^{2} $
	53	$ 1 + 276. iT - 1.48e5T^{2} $
	59	$ 1 - 212. iT - 2.05e5T^{2} $
	61	$ 1 + 174.T + 2.26e5T^{2} $
	67	$ 1 - 317.T + 3.00e5T^{2} $
	71	$ 1 - 106. iT - 3.57e5T^{2} $
	73	$ 1 + 755. iT - 3.89e5T^{2} $
	79	$ 1 - 194. iT - 4.93e5T^{2} $
	83	$ 1 + 997. iT - 5.71e5T^{2} $
	89	$ 1 - 988. iT - 7.04e5T^{2} $
	97	$ 1 + 1.02e3iT - 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

−8.771729698046327704812060111007, −8.183695045667626803686905723563, −7.45912132682957982795127991085, −6.74957986936343486981568425213, −5.65656206708707868503283461137, −4.90042193732592785820243133207, −3.60342982261052534566328446588, −2.93887489816646907674440640406, −1.63447119807662149531860102301, −0.14224223882328395862543220929, 0.62186375532973222237000479854, 2.51710931326713974428261685599, 3.69156152330246680543269635891, 4.13904925917353131932492285585, 4.99971473160399254766626791802, 6.21563337646349064363223199584, 7.24739840333891864439043724119, 7.74185347673397900584960385456, 8.608463448858285341814351543815, 9.476388551883151252398437088520

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.3682207132\)
\(L(\frac12)\)	\(\approx\)	\(0.3682207132\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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