LMFDB - L-function 2-1280-40.29-c1-0-3

Dirichlet series

L(s) = 1

− 1.70·3^-s + (−0.539 − 2.17i)5^-s − 2.63i·7^-s − 0.0783·9^-s + 5.41i·11^-s − 6.34·13^-s + (0.921 + 3.70i)15^-s − 3.41i·17^-s + 3.26i·19^-s + 4.49i·21^-s + 1.36i·23^-s + (−4.41 + 2.34i)25^-s + 5.26·27^-s − 2i·29^-s + 4.68·31^-s + ⋯

L(s) = 1

− 0.986·3^-s + (−0.241 − 0.970i)5^-s − 0.994i·7^-s − 0.0261·9^-s + 1.63i·11^-s − 1.75·13^-s + (0.237 + 0.957i)15^-s − 0.829i·17^-s + 0.748i·19^-s + 0.981i·21^-s + 0.285i·23^-s + (−0.883 + 0.468i)25^-s + 1.01·27^-s − 0.371i·29^-s + 0.840·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1280$ = $2^{8} \cdot 5$
Sign:	$0.515 - 0.856i$
Analytic conductor:	$10.2208$
Root analytic conductor:	$3.19700$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1280} (129, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1280,\ (\ :1/2),\ 0.515 - 0.856i)$

Particular Values

$L(1)$	$\approx$	$0.5199961511$
$L(\frac12)$	$\approx$	$0.5199961511$
$L(\frac{3}{2})$		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 $
	5	$ 1 + (0.539 + 2.17i)T $
good	3	$ 1 + 1.70T + 3T^{2} $
	7	$ 1 + 2.63iT - 7T^{2} $
	11	$ 1 - 5.41iT - 11T^{2} $
	13	$ 1 + 6.34T + 13T^{2} $
	17	$ 1 + 3.41iT - 17T^{2} $
	19	$ 1 - 3.26iT - 19T^{2} $
	23	$ 1 - 1.36iT - 23T^{2} $
	29	$ 1 + 2iT - 29T^{2} $
	31	$ 1 - 4.68T + 31T^{2} $
	37	$ 1 - 5.75T + 37T^{2} $
	41	$ 1 - 7.75T + 41T^{2} $
	43	$ 1 + 4.44T + 43T^{2} $
	47	$ 1 - 4.78iT - 47T^{2} $
	53	$ 1 - 1.65T + 53T^{2} $
	59	$ 1 - 3.26iT - 59T^{2} $
	61	$ 1 + 2.49iT - 61T^{2} $
	67	$ 1 + 7.86T + 67T^{2} $
	71	$ 1 + 6.15T + 71T^{2} $
	73	$ 1 - 13.5iT - 73T^{2} $
	79	$ 1 - 12.6T + 79T^{2} $
	83	$ 1 - 14.9T + 83T^{2} $
	89	$ 1 + 8.52T + 89T^{2} $
	97	$ 1 - 4.58iT - 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

−9.834693843204961011991146622852, −9.285986870856382191339497428055, −7.82529151256615138653124892891, −7.46508779119000103436632174360, −6.54207313026691912309314758857, −5.36486924657041004925599011656, −4.73055137303972126295226225128, −4.19589291258586748997681064795, −2.44835801983859669160235899421, −0.960970335086214551497487309086, 0.30876343097875368341812398357, 2.46420850472633657284426807785, 3.12795050311415123732220028360, 4.57668564658521570333875182252, 5.56232574930913587052332000547, 6.10744182528772070206250949688, 6.84317761253731196787174036594, 7.903750566549349790398683931596, 8.687719021279681988932569935160, 9.633230553684100185731541110988

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5199961511\)
\(L(\frac12)\)	\(\approx\)	\(0.5199961511\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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