LMFDB - L-function 2-768-8.5-c1-0-8

Dirichlet series

L(s) = 1

− i·3^-s + 2·7^-s − 9^-s + 4i·11^-s − 6i·13^-s + 6·17^-s − 2i·21^-s + 4·23^-s + 5·25^-s + i·27^-s − 4i·29^-s − 10·31^-s + 4·33^-s + 2i·37^-s − 6·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.755·7^-s − 0.333·9^-s + 1.20i·11^-s − 1.66i·13^-s + 1.45·17^-s − 0.436i·21^-s + 0.834·23^-s + 25^-s + 0.192i·27^-s − 0.742i·29^-s − 1.79·31^-s + 0.696·33^-s + 0.328i·37^-s − 0.960·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$768$ = $2^{8} \cdot 3$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$6.13251$
Root analytic conductor:	$2.47639$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{768} (385, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 768,\ (\ :1/2),\ 0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.56701 - 0.649076i$
$L(\frac12)$	$\approx$	$1.56701 - 0.649076i$
$L(\frac{3}{2})$		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 $
good	5	$ 1 - 5T^{2} $
	7	$ 1 - 2T + 7T^{2} $
	11	$ 1 - 4iT - 11T^{2} $
	13	$ 1 + 6iT - 13T^{2} $
	17	$ 1 - 6T + 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 - 4T + 23T^{2} $
	29	$ 1 + 4iT - 29T^{2} $
	31	$ 1 + 10T + 31T^{2} $
	37	$ 1 - 2iT - 37T^{2} $
	41	$ 1 - 2T + 41T^{2} $
	43	$ 1 + 8iT - 43T^{2} $
	47	$ 1 - 12T + 47T^{2} $
	53	$ 1 + 12iT - 53T^{2} $
	59	$ 1 - 4iT - 59T^{2} $
	61	$ 1 + 2iT - 61T^{2} $
	67	$ 1 - 4iT - 67T^{2} $
	71	$ 1 + 4T + 71T^{2} $
	73	$ 1 - 10T + 73T^{2} $
	79	$ 1 - 6T + 79T^{2} $
	83	$ 1 - 12iT - 83T^{2} $
	89	$ 1 + 2T + 89T^{2} $
	97	$ 1 + 6T + 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

−10.32746818568192312839748502723, −9.368856051707355902230072640025, −8.275147052231148820384204512104, −7.62636888582029667729632876830, −6.98167379803965498680802887920, −5.56960999106934779616024988406, −5.08102881373563885577068223802, −3.61326847361635129245552726008, −2.38832576738429158256579444906, −1.05358884779538257442246104043, 1.36953664321006418648610239834, 2.99890171503348173611991854582, 4.01758698833620399433183443014, 5.03844313933873952201198379022, 5.83696971187077453581934174752, 6.99709311734244714856378557170, 7.937663423525615724717844559466, 8.980194246803884730916866604967, 9.305349933171789530809623443936, 10.71952455842196254021447580282

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.56701 - 0.649076i\)
\(L(\frac12)\)	\(\approx\)	\(1.56701 - 0.649076i\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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