LMFDB - L-function 2-1280-40.19-c2-0-66

Dirichlet series

L(s) = 1

− 2.82i·3^-s + (−4.89 − i)5^-s − 8.48·7^-s + 0.999·9^-s + 13.8·11^-s + 9.79·13^-s + (−2.82 + 13.8i)15^-s − 19.5i·17^-s + 13.8·19^-s + 24i·21^-s + 25.4·23^-s + (22.9 + 9.79i)25^-s − 28.2i·27^-s + 22i·29^-s − 55.4i·31^-s + ⋯

L(s) = 1

− 0.942i·3^-s + (−0.979 − 0.200i)5^-s − 1.21·7^-s + 0.111·9^-s + 1.25·11^-s + 0.753·13^-s + (−0.188 + 0.923i)15^-s − 1.15i·17^-s + 0.729·19^-s + 1.14i·21^-s + 1.10·23^-s + (0.919 + 0.391i)25^-s − 1.04i·27^-s + 0.758i·29^-s − 1.78i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1280$ = $2^{8} \cdot 5$
Sign:	$-0.834 + 0.551i$
Analytic conductor:	$34.8774$
Root analytic conductor:	$5.90571$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1280} (639, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1280,\ (\ :1),\ -0.834 + 0.551i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.268990802$
$L(\frac12)$	$\approx$	$1.268990802$
$L(2)$		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 $
	5	$ 1 + (4.89 + i)T $
good	3	$ 1 + 2.82iT - 9T^{2} $
	7	$ 1 + 8.48T + 49T^{2} $
	11	$ 1 - 13.8T + 121T^{2} $
	13	$ 1 - 9.79T + 169T^{2} $
	17	$ 1 + 19.5iT - 289T^{2} $
	19	$ 1 - 13.8T + 361T^{2} $
	23	$ 1 - 25.4T + 529T^{2} $
	29	$ 1 - 22iT - 841T^{2} $
	31	$ 1 + 55.4iT - 961T^{2} $
	37	$ 1 + 48.9T + 1.36e3T^{2} $
	41	$ 1 + 22T + 1.68e3T^{2} $
	43	$ 1 - 59.3iT - 1.84e3T^{2} $
	47	$ 1 - 8.48T + 2.20e3T^{2} $
	53	$ 1 + 29.3T + 2.80e3T^{2} $
	59	$ 1 - 13.8T + 3.48e3T^{2} $
	61	$ 1 + 46iT - 3.72e3T^{2} $
	67	$ 1 + 59.3iT - 4.48e3T^{2} $
	71	$ 1 + 27.7iT - 5.04e3T^{2} $
	73	$ 1 + 78.3iT - 5.32e3T^{2} $
	79	$ 1 - 6.24e3T^{2} $
	83	$ 1 - 76.3iT - 6.88e3T^{2} $
	89	$ 1 + 146T + 7.92e3T^{2} $
	97	$ 1 + 58.7iT - 9.40e3T^{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

−9.215203697259331032011608463229, −8.242639311453951256304002608582, −7.28132397018448303557854593325, −6.86024647043114197254453792644, −6.13217462680165844553381517619, −4.81184383749242090248764460507, −3.72304865709449217570900730852, −3.03327583626180496754828739314, −1.37816848971554490851505166497, −0.44756476308975170954149282885, 1.23029604134116961306875750339, 3.30348922753349584337643748588, 3.60825974129404079908963242275, 4.41005701523569670498121603019, 5.57950546742194049777445018667, 6.74320942958237330449946785975, 7.05942250165022912534125919976, 8.542275470011272314563004964739, 8.939959949170675392215777110523, 9.889057233501706984582117826167

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.268990802\)
\(L(\frac12)\)	\(\approx\)	\(1.268990802\)
\(L(2)\)		not available
\(L(1)\)		not available

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