LMFDB - L-function 2-600-5.4-c3-0-12

Dirichlet series

L(s) = 1

− 3i·3^-s − 21.8i·7^-s − 9·9^-s + 54.6·11^-s + 82.7i·13^-s + 100. i·17^-s + 84.1·19^-s − 65.6·21^-s − 0.880i·23^-s + 27i·27^-s + 99.1·29^-s − 78.9·31^-s − 163. i·33^-s − 390. i·37^-s + 248.·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 1.18i·7^-s − 0.333·9^-s + 1.49·11^-s + 1.76i·13^-s + 1.43i·17^-s + 1.01·19^-s − 0.682·21^-s − 0.00798i·23^-s + 0.192i·27^-s + 0.634·29^-s − 0.457·31^-s − 0.864i·33^-s − 1.73i·37^-s + 1.01·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$600$ = $2^{3} \cdot 3 \cdot 5^{2}$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$35.4011$
Root analytic conductor:	$5.94988$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{600} (49, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 600,\ (\ :3/2),\ 0.894 + 0.447i)$

Particular Values

$L(2)$	$\approx$	$2.209170534$
$L(\frac12)$	$\approx$	$2.209170534$
$L(\frac{5}{2})$		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 + 21.8iT - 343T^{2} $
	11	$ 1 - 54.6T + 1.33e3T^{2} $
	13	$ 1 - 82.7iT - 2.19e3T^{2} $
	17	$ 1 - 100. iT - 4.91e3T^{2} $
	19	$ 1 - 84.1T + 6.85e3T^{2} $
	23	$ 1 + 0.880iT - 1.21e4T^{2} $
	29	$ 1 - 99.1T + 2.43e4T^{2} $
	31	$ 1 + 78.9T + 2.97e4T^{2} $
	37	$ 1 + 390. iT - 5.06e4T^{2} $
	41	$ 1 - 104.T + 6.89e4T^{2} $
	43	$ 1 - 241. iT - 7.95e4T^{2} $
	47	$ 1 + 512. iT - 1.03e5T^{2} $
	53	$ 1 - 284. iT - 1.48e5T^{2} $
	59	$ 1 - 709.T + 2.05e5T^{2} $
	61	$ 1 - 470.T + 2.26e5T^{2} $
	67	$ 1 - 667. iT - 3.00e5T^{2} $
	71	$ 1 + 51.5T + 3.57e5T^{2} $
	73	$ 1 - 371. iT - 3.89e5T^{2} $
	79	$ 1 - 79.3T + 4.93e5T^{2} $
	83	$ 1 + 682. iT - 5.71e5T^{2} $
	89	$ 1 + 628.T + 7.04e5T^{2} $
	97	$ 1 + 1.51e3iT - 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

−10.18213166203725245969871762292, −9.260916722674592424919815561531, −8.485438256903459492112235796254, −7.21175957323481055115377735442, −6.84859506167256580968200480595, −5.85782973887481707375935130866, −4.25103008060284159014230339010, −3.75117838340060833834142103517, −1.88443403498103282605214314525, −0.984032761688385399213954261270, 0.891644278167567419717535854652, 2.66838639305573628472490652511, 3.48927439138251199564886102906, 4.93042556554857431343952765600, 5.57892703307527982926781662397, 6.60912453406999823197502424841, 7.81053311824142499223817108118, 8.764338299455004412145711589565, 9.449499140256462686613694399912, 10.13253668542405563045392012195

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 $\F_{q}$
Belyi maps

\(L(2)\)	\(\approx\)	\(2.209170534\)
\(L(\frac12)\)	\(\approx\)	\(2.209170534\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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