LMFDB - L-function 2-1040-65.64-c1-0-30

Dirichlet series

L(s) = 1

− 2.52i·3^-s + (2.18 + 0.469i)5^-s + 2.37·7^-s − 3.37·9^-s − 1.58i·11^-s + (1 + 3.46i)13^-s + (1.18 − 5.51i)15^-s − 5.98i·17^-s − 3.46i·19^-s − 5.98i·21^-s + 6.63i·23^-s + (4.55 + 2.05i)25^-s + 0.939i·27^-s + 2.74·29^-s + 3.46i·31^-s + ⋯

L(s) = 1

− 1.45i·3^-s + (0.977 + 0.210i)5^-s + 0.896·7^-s − 1.12·9^-s − 0.477i·11^-s + (0.277 + 0.960i)13^-s + (0.306 − 1.42i)15^-s − 1.45i·17^-s − 0.794i·19^-s − 1.30i·21^-s + 1.38i·23^-s + (0.911 + 0.410i)25^-s + 0.180i·27^-s + 0.509·29^-s + 0.622i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0692 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0692 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1040$ = $2^{4} \cdot 5 \cdot 13$
Sign:	$0.0692 + 0.997i$
Analytic conductor:	$8.30444$
Root analytic conductor:	$2.88174$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1040} (129, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1040,\ (\ :1/2),\ 0.0692 + 0.997i)$

Particular Values

$L(1)$	$\approx$	$2.131678663$
$L(\frac12)$	$\approx$	$2.131678663$
$L(\frac{3}{2})$		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 + (-2.18 - 0.469i)T $
	13	$ 1 + (-1 - 3.46i)T $
good	3	$ 1 + 2.52iT - 3T^{2} $
	7	$ 1 - 2.37T + 7T^{2} $
	11	$ 1 + 1.58iT - 11T^{2} $
	17	$ 1 + 5.98iT - 17T^{2} $
	19	$ 1 + 3.46iT - 19T^{2} $
	23	$ 1 - 6.63iT - 23T^{2} $
	29	$ 1 - 2.74T + 29T^{2} $
	31	$ 1 - 3.46iT - 31T^{2} $
	37	$ 1 - 9.11T + 37T^{2} $
	41	$ 1 + 10.0iT - 41T^{2} $
	43	$ 1 + 0.644iT - 43T^{2} $
	47	$ 1 + 10.3T + 47T^{2} $
	53	$ 1 - 5.04iT - 53T^{2} $
	59	$ 1 - 6.63iT - 59T^{2} $
	61	$ 1 + 6.74T + 61T^{2} $
	67	$ 1 - 4T + 67T^{2} $
	71	$ 1 + 12.6iT - 71T^{2} $
	73	$ 1 + 10T + 73T^{2} $
	79	$ 1 + 4.74T + 79T^{2} $
	83	$ 1 - 8.74T + 83T^{2} $
	89	$ 1 + 1.87iT - 89T^{2} $
	97	$ 1 + 6.74T + 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

−9.478596640377634202833244913991, −8.921173354974019920885424456395, −7.86886699455748704335924096095, −7.15232948683292260897159252513, −6.50322479138415080886145672269, −5.60728989479020817247624891464, −4.69993356775373143708741364011, −2.96044689980600488887318070184, −1.98325551657214501191908861200, −1.11085236737372056363683589779, 1.57502306626708735021521025029, 2.91871872017111522801500133503, 4.20053911965770776774727217881, 4.78106693676393916024196587797, 5.68574521851644921912655584642, 6.41320598889990220223492505210, 8.139653701198935307240628933274, 8.389759602316989574914249730107, 9.644809833255809789799790038508, 10.02345103386897597912060878739

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.131678663\)
\(L(\frac12)\)	\(\approx\)	\(2.131678663\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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