LMFDB - L-function 2-3024-21.20-c1-0-21

Dirichlet series

L(s) = 1

− 0.933·5^-s + (−2.45 + 0.981i)7^-s − 1.75i·11^-s − 4.20i·13^-s + 0.231·17^-s + 5.00i·19^-s + 2.61i·23^-s − 4.12·25^-s + 3.77i·29^-s + 0.840i·31^-s + (2.29 − 0.915i)35^-s + 3.01·37^-s + 8.98·41^-s − 2.40·43^-s + 1.03·47^-s + ⋯

L(s) = 1

− 0.417·5^-s + (−0.928 + 0.371i)7^-s − 0.529i·11^-s − 1.16i·13^-s + 0.0560·17^-s + 1.14i·19^-s + 0.545i·23^-s − 0.825·25^-s + 0.700i·29^-s + 0.150i·31^-s + (0.387 − 0.154i)35^-s + 0.495·37^-s + 1.40·41^-s − 0.366·43^-s + 0.150·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3024$ = $2^{4} \cdot 3^{3} \cdot 7$
Sign:	$0.928 - 0.371i$
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:	$(2,\ 3024,\ (\ :1/2),\ 0.928 - 0.371i)$

Particular Values

$L(1)$	$\approx$	$1.224458223$
$L(\frac12)$	$\approx$	$1.224458223$
$L(\frac{3}{2})$		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 + (2.45 - 0.981i)T $
good	5	$ 1 + 0.933T + 5T^{2} $
	11	$ 1 + 1.75iT - 11T^{2} $
	13	$ 1 + 4.20iT - 13T^{2} $
	17	$ 1 - 0.231T + 17T^{2} $
	19	$ 1 - 5.00iT - 19T^{2} $
	23	$ 1 - 2.61iT - 23T^{2} $
	29	$ 1 - 3.77iT - 29T^{2} $
	31	$ 1 - 0.840iT - 31T^{2} $
	37	$ 1 - 3.01T + 37T^{2} $
	41	$ 1 - 8.98T + 41T^{2} $
	43	$ 1 + 2.40T + 43T^{2} $
	47	$ 1 - 1.03T + 47T^{2} $
	53	$ 1 + 9.04iT - 53T^{2} $
	59	$ 1 - 6.77T + 59T^{2} $
	61	$ 1 + 8.93iT - 61T^{2} $
	67	$ 1 + 1.10T + 67T^{2} $
	71	$ 1 - 7.21iT - 71T^{2} $
	73	$ 1 + 3.44iT - 73T^{2} $
	79	$ 1 + 5.29T + 79T^{2} $
	83	$ 1 - 14.8T + 83T^{2} $
	89	$ 1 - 15.0T + 89T^{2} $
	97	$ 1 + 1.63iT - 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

−8.708563317535161465228892072384, −7.981460544069878736482291268248, −7.42778677788262279962702782714, −6.33267142810182078058007138878, −5.81906397713802163100491728523, −5.04909884154934673196488343104, −3.68613113591769369611489127076, −3.38929885864248993989306252575, −2.24654137567586051453626169011, −0.74411759913879304394623141433, 0.57921180230010721327491015717, 2.10452651003610476733163997784, 3.01693247426164675270512853083, 4.18850977898720852494340523486, 4.43349138546754163781532802373, 5.76811148457698631441290942021, 6.51795050879697183098578794686, 7.16533704291169369067847764340, 7.76479778679911378712428561116, 8.840835763970725298160708987993

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.224458223\)
\(L(\frac12)\)	\(\approx\)	\(1.224458223\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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