LMFDB - L-function 2-1200-5.4-c3-0-19

Dirichlet series

L(s) = 1

− 3i·3^-s − 4i·7^-s − 9·9^-s + 48·11^-s + 2i·13^-s + 114i·17^-s + 140·19^-s − 12·21^-s − 72i·23^-s + 27i·27^-s − 210·29^-s − 272·31^-s − 144i·33^-s + 334i·37^-s + 6·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 0.215i·7^-s − 0.333·9^-s + 1.31·11^-s + 0.0426i·13^-s + 1.62i·17^-s + 1.69·19^-s − 0.124·21^-s − 0.652i·23^-s + 0.192i·27^-s − 1.34·29^-s − 1.57·31^-s − 0.759i·33^-s + 1.48i·37^-s + 0.0246·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1200$ = $2^{4} \cdot 3 \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$70.8022$
Root analytic conductor:	$8.41440$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1200} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1200,\ (\ :3/2),\ 0.894 - 0.447i)$

Particular Values

$L(2)$	$\approx$	$2.063733399$
$L(\frac12)$	$\approx$	$2.063733399$
$L(\frac{5}{2})$		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 + 3iT $
	5	$ 1 $
good	7	$ 1 + 4iT - 343T^{2} $
	11	$ 1 - 48T + 1.33e3T^{2} $
	13	$ 1 - 2iT - 2.19e3T^{2} $
	17	$ 1 - 114iT - 4.91e3T^{2} $
	19	$ 1 - 140T + 6.85e3T^{2} $
	23	$ 1 + 72iT - 1.21e4T^{2} $
	29	$ 1 + 210T + 2.43e4T^{2} $
	31	$ 1 + 272T + 2.97e4T^{2} $
	37	$ 1 - 334iT - 5.06e4T^{2} $
	41	$ 1 + 198T + 6.89e4T^{2} $
	43	$ 1 - 268iT - 7.95e4T^{2} $
	47	$ 1 - 216iT - 1.03e5T^{2} $
	53	$ 1 + 78iT - 1.48e5T^{2} $
	59	$ 1 - 240T + 2.05e5T^{2} $
	61	$ 1 - 302T + 2.26e5T^{2} $
	67	$ 1 - 596iT - 3.00e5T^{2} $
	71	$ 1 - 768T + 3.57e5T^{2} $
	73	$ 1 + 478iT - 3.89e5T^{2} $
	79	$ 1 + 640T + 4.93e5T^{2} $
	83	$ 1 - 348iT - 5.71e5T^{2} $
	89	$ 1 + 210T + 7.04e5T^{2} $
	97	$ 1 - 1.53e3iT - 9.12e5T^{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

−9.359799366833261840636254597388, −8.605876664799966058516507915818, −7.72943548797226142481689761762, −6.94910197208657226420546753877, −6.19560886629945639039652644543, −5.34628656762851323706343060284, −4.05721486601111057134018429552, −3.32786197715328077176298711000, −1.83957999922518805068485831117, −1.05292629844596158041362267066, 0.56398853839584598481768331207, 1.94103194222554993707145549004, 3.31210587949333384787610363608, 3.92338759499866711659654893239, 5.23391451551795439624459672109, 5.63641496991596172262758155199, 7.05710802281960569950249482998, 7.44456820201042208779106791273, 8.871559736343740868445159615325, 9.312330133089882012630771800149

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.063733399\)
\(L(\frac12)\)	\(\approx\)	\(2.063733399\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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