LMFDB - L-function 2-3024-21.20-c1-0-37

Dirichlet series

L(s) = 1

− 1.73·5^-s + (−1 − 2.44i)7^-s − 1.41i·11^-s + 2.44i·13^-s + 5.19·17^-s + 7.34i·19^-s + 2.82i·23^-s − 2.00·25^-s − 7.07i·29^-s − 2.44i·31^-s + (1.73 + 4.24i)35^-s + 5·37^-s + 8.66·41^-s − 5·43^-s + 8.66·47^-s + ⋯

L(s) = 1

− 0.774·5^-s + (−0.377 − 0.925i)7^-s − 0.426i·11^-s + 0.679i·13^-s + 1.26·17^-s + 1.68i·19^-s + 0.589i·23^-s − 0.400·25^-s − 1.31i·29^-s − 0.439i·31^-s + (0.292 + 0.717i)35^-s + 0.821·37^-s + 1.35·41^-s − 0.762·43^-s + 1.26·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3024$ = $2^{4} \cdot 3^{3} \cdot 7$
Sign:	$0.377 + 0.925i$
Analytic conductor:	$24.1467$
Root analytic conductor:	$4.91393$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3024} (1889, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3024,\ (\ :1/2),\ 0.377 + 0.925i)$

Particular Values

$L(1)$	$\approx$	$1.241360044$
$L(\frac12)$	$\approx$	$1.241360044$
$L(\frac{3}{2})$		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 $
	7	$ 1 + (1 + 2.44i)T $
good	5	$ 1 + 1.73T + 5T^{2} $
	11	$ 1 + 1.41iT - 11T^{2} $
	13	$ 1 - 2.44iT - 13T^{2} $
	17	$ 1 - 5.19T + 17T^{2} $
	19	$ 1 - 7.34iT - 19T^{2} $
	23	$ 1 - 2.82iT - 23T^{2} $
	29	$ 1 + 7.07iT - 29T^{2} $
	31	$ 1 + 2.44iT - 31T^{2} $
	37	$ 1 - 5T + 37T^{2} $
	41	$ 1 - 8.66T + 41T^{2} $
	43	$ 1 + 5T + 43T^{2} $
	47	$ 1 - 8.66T + 47T^{2} $
	53	$ 1 + 11.3iT - 53T^{2} $
	59	$ 1 + 8.66T + 59T^{2} $
	61	$ 1 + 2.44iT - 61T^{2} $
	67	$ 1 + 2T + 67T^{2} $
	71	$ 1 + 14.1iT - 71T^{2} $
	73	$ 1 - 73T^{2} $
	79	$ 1 - 13T + 79T^{2} $
	83	$ 1 + 1.73T + 83T^{2} $
	89	$ 1 + 10.3T + 89T^{2} $
	97	$ 1 + 17.1iT - 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

−8.305467701370379544618459082287, −7.78760632315514399560430889462, −7.33173853232825300273092888702, −6.21397935896695002107825542070, −5.71489816820015842887008912033, −4.39397653356963753801259640463, −3.85211206974430823711303568972, −3.19701259060450541425572749375, −1.71976668654173784339913194175, −0.50176122247856253311475649149, 0.946872340514202530897716208534, 2.53391645658292052249275060187, 3.11923114137449066387456638536, 4.16310535481772953083875700005, 5.06758980826642576927217905432, 5.73531807293022851674778658168, 6.66765223256726731678443406938, 7.46239140112011515492787183628, 8.040633312890816906626301760062, 8.952491534193069840731753478085

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.241360044\)
\(L(\frac12)\)	\(\approx\)	\(1.241360044\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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