LMFDB - L-function 2-3360-140.139-c1-0-81

Dirichlet series

L(s) = 1

− i·3^-s + (−2.00 + 0.995i)5^-s + (−1.37 − 2.25i)7^-s − 9^-s − 3.02i·11^-s + 1.42·13^-s + (0.995 + 2.00i)15^-s + 7.83·17^-s + 0.687·19^-s + (−2.25 + 1.37i)21^-s − 4.79·23^-s + (3.01 − 3.98i)25^-s + i·27^-s + 7.65·29^-s − 10.5·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + (−0.895 + 0.445i)5^-s + (−0.521 − 0.853i)7^-s − 0.333·9^-s − 0.912i·11^-s + 0.396·13^-s + (0.257 + 0.516i)15^-s + 1.90·17^-s + 0.157·19^-s + (−0.492 + 0.300i)21^-s − 1.00·23^-s + (0.603 − 0.797i)25^-s + 0.192i·27^-s + 1.42·29^-s − 1.89·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3360$ = $2^{5} \cdot 3 \cdot 5 \cdot 7$
Sign:	$-0.974 + 0.222i$
Analytic conductor:	$26.8297$
Root analytic conductor:	$5.17974$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3360} (2239, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3360,\ (\ :1/2),\ -0.974 + 0.222i)$

Particular Values

$L(1)$	$\approx$	$0.8111206578$
$L(\frac12)$	$\approx$	$0.8111206578$
$L(\frac{3}{2})$		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 + iT $
	5	$ 1 + (2.00 - 0.995i)T $
	7	$ 1 + (1.37 + 2.25i)T $
good	11	$ 1 + 3.02iT - 11T^{2} $
	13	$ 1 - 1.42T + 13T^{2} $
	17	$ 1 - 7.83T + 17T^{2} $
	19	$ 1 - 0.687T + 19T^{2} $
	23	$ 1 + 4.79T + 23T^{2} $
	29	$ 1 - 7.65T + 29T^{2} $
	31	$ 1 + 10.5T + 31T^{2} $
	37	$ 1 + 8.43iT - 37T^{2} $
	41	$ 1 - 5.26iT - 41T^{2} $
	43	$ 1 - 7.78T + 43T^{2} $
	47	$ 1 + 5.98iT - 47T^{2} $
	53	$ 1 + 11.6iT - 53T^{2} $
	59	$ 1 + 8.22T + 59T^{2} $
	61	$ 1 - 13.7iT - 61T^{2} $
	67	$ 1 - 2.90T + 67T^{2} $
	71	$ 1 - 0.334iT - 71T^{2} $
	73	$ 1 + 2.35T + 73T^{2} $
	79	$ 1 + 2.84iT - 79T^{2} $
	83	$ 1 + 11.1iT - 83T^{2} $
	89	$ 1 - 1.41iT - 89T^{2} $
	97	$ 1 + 18.4T + 97T^{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

−8.012009218665175245777172875260, −7.59560266400784391908852967878, −6.91847006979658668001533837049, −6.09648010889312651013892423348, −5.43903182201939705181455711973, −4.05185703158253466174052202548, −3.55007218260667761829848618248, −2.81137700832521373258357483984, −1.25363859970818072155309843867, −0.28179685565234204827296915239, 1.33774687153073933480848344786, 2.74969775903540732057624170935, 3.54164458049610888265288587216, 4.27035384321207205292468397604, 5.18774356336934993308956099116, 5.75660152926749687843360486970, 6.72078869180027777148600724571, 7.75004788884199560598194685008, 8.115433124218350893972037759239, 9.114348042419081017688510445197

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8111206578\)
\(L(\frac12)\)	\(\approx\)	\(0.8111206578\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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