LMFDB - L-function 2-960-5.4-c3-0-40

Dirichlet series

L(s) = 1

− 3i·3^-s + (−2 + 11i)5^-s − 2i·7^-s − 9·9^-s + 70·11^-s − 54i·13^-s + (33 + 6i)15^-s + 22i·17^-s − 24·19^-s − 6·21^-s + 100i·23^-s + (−117 − 44i)25^-s + 27i·27^-s + 216·29^-s − 208·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + (−0.178 + 0.983i)5^-s − 0.107i·7^-s − 0.333·9^-s + 1.91·11^-s − 1.15i·13^-s + (0.568 + 0.103i)15^-s + 0.313i·17^-s − 0.289·19^-s − 0.0623·21^-s + 0.906i·23^-s + (−0.936 − 0.351i)25^-s + 0.192i·27^-s + 1.38·29^-s − 1.20·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$960$ = $2^{6} \cdot 3 \cdot 5$
Sign:	$0.983 + 0.178i$
Analytic conductor:	$56.6418$
Root analytic conductor:	$7.52607$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{960} (769, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 960,\ (\ :3/2),\ 0.983 + 0.178i)$

Particular Values

$L(2)$	$\approx$	$2.178599532$
$L(\frac12)$	$\approx$	$2.178599532$
$L(\frac{5}{2})$		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 $
	5	$ 1 + (2 - 11i)T $
good	7	$ 1 + 2iT - 343T^{2} $
	11	$ 1 - 70T + 1.33e3T^{2} $
	13	$ 1 + 54iT - 2.19e3T^{2} $
	17	$ 1 - 22iT - 4.91e3T^{2} $
	19	$ 1 + 24T + 6.85e3T^{2} $
	23	$ 1 - 100iT - 1.21e4T^{2} $
	29	$ 1 - 216T + 2.43e4T^{2} $
	31	$ 1 + 208T + 2.97e4T^{2} $
	37	$ 1 + 254iT - 5.06e4T^{2} $
	41	$ 1 + 206T + 6.89e4T^{2} $
	43	$ 1 - 292iT - 7.95e4T^{2} $
	47	$ 1 + 320iT - 1.03e5T^{2} $
	53	$ 1 - 402iT - 1.48e5T^{2} $
	59	$ 1 - 370T + 2.05e5T^{2} $
	61	$ 1 - 550T + 2.26e5T^{2} $
	67	$ 1 + 728iT - 3.00e5T^{2} $
	71	$ 1 - 540T + 3.57e5T^{2} $
	73	$ 1 - 604iT - 3.89e5T^{2} $
	79	$ 1 - 792T + 4.93e5T^{2} $
	83	$ 1 - 404iT - 5.71e5T^{2} $
	89	$ 1 - 938T + 7.04e5T^{2} $
	97	$ 1 + 56iT - 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

−9.640015039089004751355671340319, −8.717425008929317773583590780597, −7.81333637005420609414859209230, −6.99164220618287742172170881065, −6.39808282576933830047452847566, −5.51420709992944018050773008002, −3.98150145691963405328867845134, −3.29874965429420689782219854862, −2.02981178325438252640846382049, −0.808528621866258996298362676715, 0.828121072329971548702486123571, 1.99198318130975708246336157194, 3.65898375533114537101263828948, 4.31465635229975468909707521738, 5.07533428766961604031997568931, 6.30600560368817961417891502801, 6.95119104737054049554836208855, 8.382032193974636905220141939523, 8.916077389744112995404585936578, 9.443027152839833840281802098872

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.178599532\)
\(L(\frac12)\)	\(\approx\)	\(2.178599532\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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