LMFDB - L-function 2-6012-501.500-c1-0-17

Dirichlet series

L(s) = 1

− 3.04·5^-s + 0.233·7^-s − 2.11i·11^-s + 1.71i·13^-s + 5.18·17^-s − 1.94·19^-s − 2.99·23^-s + 4.26·25^-s − 2.73i·29^-s − 8.50·31^-s − 0.711·35^-s + 8.21i·37^-s + 3.33·41^-s − 0.222i·43^-s − 8.47i·47^-s + ⋯

L(s) = 1

− 1.36·5^-s + 0.0883·7^-s − 0.636i·11^-s + 0.475i·13^-s + 1.25·17^-s − 0.446·19^-s − 0.624·23^-s + 0.852·25^-s − 0.508i·29^-s − 1.52·31^-s − 0.120·35^-s + 1.35i·37^-s + 0.521·41^-s − 0.0339i·43^-s − 1.23i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6012$ = $2^{2} \cdot 3^{2} \cdot 167$
Sign:	$0.681 - 0.731i$
Analytic conductor:	$48.0060$
Root analytic conductor:	$6.92864$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6012} (3005, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6012,\ (\ :1/2),\ 0.681 - 0.731i)$

Particular Values

$L(1)$	$\approx$	$1.021130896$
$L(\frac12)$	$\approx$	$1.021130896$
$L(\frac{3}{2})$		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 $
	167	$ 1 + (12.8 + 1.73i)T $
good	5	$ 1 + 3.04T + 5T^{2} $
	7	$ 1 - 0.233T + 7T^{2} $
	11	$ 1 + 2.11iT - 11T^{2} $
	13	$ 1 - 1.71iT - 13T^{2} $
	17	$ 1 - 5.18T + 17T^{2} $
	19	$ 1 + 1.94T + 19T^{2} $
	23	$ 1 + 2.99T + 23T^{2} $
	29	$ 1 + 2.73iT - 29T^{2} $
	31	$ 1 + 8.50T + 31T^{2} $
	37	$ 1 - 8.21iT - 37T^{2} $
	41	$ 1 - 3.33T + 41T^{2} $
	43	$ 1 + 0.222iT - 43T^{2} $
	47	$ 1 + 8.47iT - 47T^{2} $
	53	$ 1 - 9.15T + 53T^{2} $
	59	$ 1 + 10.8T + 59T^{2} $
	61	$ 1 - 10.7T + 61T^{2} $
	67	$ 1 + 0.936iT - 67T^{2} $
	71	$ 1 - 9.56T + 71T^{2} $
	73	$ 1 - 9.59iT - 73T^{2} $
	79	$ 1 + 5.20iT - 79T^{2} $
	83	$ 1 + 3.37T + 83T^{2} $
	89	$ 1 + 16.1iT - 89T^{2} $
	97	$ 1 + 11.2T + 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

−8.211994690735086968383718690468, −7.53036790491208773360177300369, −6.92274853834518038274440769457, −6.02090922380350949014852013925, −5.30150686408574968356845531140, −4.37634207116122842124345642182, −3.73200269264011270049439979739, −3.17133644285313669566302826811, −1.94185765751869114757334712307, −0.71248072178876367125046208223, 0.39663182021784931307944402585, 1.65345016156051896947691910799, 2.80543880473554331515074568789, 3.72885242089616914723295073762, 4.11702951056827964894257747220, 5.11280279368721711509071841770, 5.73723224737375034232799566057, 6.74779454796225873892652112943, 7.56450727233622823571458129391, 7.76167523827519319153751900514

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.021130896\)
\(L(\frac12)\)	\(\approx\)	\(1.021130896\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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