LMFDB - L-function 2-1200-60.59-c1-0-2

Dirichlet series

L(s) = 1

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

L(s) = 1

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0599 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0599 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1200$ = $2^{4} \cdot 3 \cdot 5^{2}$
Sign:	$-0.0599 - 0.998i$
Analytic conductor:	$9.58204$
Root analytic conductor:	$3.09548$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1200} (1199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1200,\ (\ :1/2),\ -0.0599 - 0.998i)$

Particular Values

$L(1)$	$\approx$	$0.3227412984$
$L(\frac12)$	$\approx$	$0.3227412984$
$L(\frac{3}{2})$		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 + 1.73iT $
	5	$ 1 $
good	7	$ 1 + 3.46T + 7T^{2} $
	11	$ 1 + 3.46T + 11T^{2} $
	13	$ 1 + 4iT - 13T^{2} $
	17	$ 1 - 6T + 17T^{2} $
	19	$ 1 - 3.46iT - 19T^{2} $
	23	$ 1 - 3.46iT - 23T^{2} $
	29	$ 1 - 6iT - 29T^{2} $
	31	$ 1 - 3.46iT - 31T^{2} $
	37	$ 1 + 4iT - 37T^{2} $
	41	$ 1 - 12iT - 41T^{2} $
	43	$ 1 + 6.92T + 43T^{2} $
	47	$ 1 - 3.46iT - 47T^{2} $
	53	$ 1 + 6T + 53T^{2} $
	59	$ 1 - 3.46T + 59T^{2} $
	61	$ 1 + 10T + 61T^{2} $
	67	$ 1 - 6.92T + 67T^{2} $
	71	$ 1 + 13.8T + 71T^{2} $
	73	$ 1 - 2iT - 73T^{2} $
	79	$ 1 + 10.3iT - 79T^{2} $
	83	$ 1 + 10.3iT - 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 - 10iT - 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

−10.06078838463577685714102350712, −9.132984196246090727179682120483, −7.960274382343062594505721771394, −7.69468976842032500805903563471, −6.63064684602502409705402700515, −5.82147497366625092112464639473, −5.21391367531390661488275909399, −3.26070601934109285016083911164, −3.02466625771428007346740896713, −1.35616875253277572459387037971, 0.13913519217452760597168849125, 2.49288223119239946990374059845, 3.33557797184952633163442298521, 4.26529821317504420513894404820, 5.25520674165005389280692451914, 6.07309799562132128577814309949, 6.96153231702213991202372303889, 8.040859998437126273922459313723, 8.923950932926339665345100251167, 9.751097024153991486737584027478

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.3227412984\)
\(L(\frac12)\)	\(\approx\)	\(0.3227412984\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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