LMFDB - L-function 2-1200-12.11-c1-0-22

Dirichlet series

L(s) = 1

− 1.73·3^-s + 3.46i·7^-s + 2.99·9^-s − 3.46·11^-s − 4·13^-s − 6i·17^-s + 3.46i·19^-s − 5.99i·21^-s + 3.46·23^-s − 5.19·27^-s + 6i·29^-s − 3.46i·31^-s + 5.99·33^-s + 4·37^-s + 6.92·39^-s + ⋯

L(s) = 1

− 1.00·3^-s + 1.30i·7^-s + 0.999·9^-s − 1.04·11^-s − 1.10·13^-s − 1.45i·17^-s + 0.794i·19^-s − 1.30i·21^-s + 0.722·23^-s − 1.00·27^-s + 1.11i·29^-s − 0.622i·31^-s + 1.04·33^-s + 0.657·37^-s + 1.10·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1200$ = $2^{4} \cdot 3 \cdot 5^{2}$
Sign:	$-0.5 + 0.866i$
Analytic conductor:	$9.58204$
Root analytic conductor:	$3.09548$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1200} (1151, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1200,\ (\ :1/2),\ -0.5 + 0.866i)$

Particular Values

$L(1)$	$\approx$	$0.2812116991$
$L(\frac12)$	$\approx$	$0.2812116991$
$L(\frac{3}{2})$		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 + 1.73T $
	5	$ 1 $
good	7	$ 1 - 3.46iT - 7T^{2} $
	11	$ 1 + 3.46T + 11T^{2} $
	13	$ 1 + 4T + 13T^{2} $
	17	$ 1 + 6iT - 17T^{2} $
	19	$ 1 - 3.46iT - 19T^{2} $
	23	$ 1 - 3.46T + 23T^{2} $
	29	$ 1 - 6iT - 29T^{2} $
	31	$ 1 + 3.46iT - 31T^{2} $
	37	$ 1 - 4T + 37T^{2} $
	41	$ 1 + 12iT - 41T^{2} $
	43	$ 1 + 6.92iT - 43T^{2} $
	47	$ 1 + 3.46T + 47T^{2} $
	53	$ 1 + 6iT - 53T^{2} $
	59	$ 1 + 3.46T + 59T^{2} $
	61	$ 1 + 10T + 61T^{2} $
	67	$ 1 + 6.92iT - 67T^{2} $
	71	$ 1 + 13.8T + 71T^{2} $
	73	$ 1 - 2T + 73T^{2} $
	79	$ 1 + 10.3iT - 79T^{2} $
	83	$ 1 + 10.3T + 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 + 10T + 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

−9.559154992475710805620485588468, −8.804330476792746120025822901236, −7.60142384595202440677427650662, −7.05813651340714620350130199679, −5.84149255384127535926568052636, −5.29820230688059406651349201561, −4.68128785864173441191328753785, −3.04854332335338325993964436251, −2.05035438238152100739931983970, −0.14495271801349587772582375727, 1.24949974391807442269895498300, 2.84981593718342395936760095328, 4.30818213461729069166455355256, 4.75222893608096037942303067435, 5.85155468172535609992734301066, 6.71866854605241423728366863602, 7.48471459619157969804729289078, 8.093903893531448088581522837653, 9.550851924248552049879894815737, 10.17961646060576290664693340363

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.2812116991\)
\(L(\frac12)\)	\(\approx\)	\(0.2812116991\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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