LMFDB - L-function 2-6048-12.11-c1-0-19

Dirichlet series

L(s) = 1

− 2.21i·5^-s − i·7^-s − 0.614·11^-s − 5.56·13^-s + 0.585i·17^-s + 3.89i·19^-s − 0.850·23^-s + 0.0963·25^-s + 4.74i·29^-s − 3.07i·31^-s − 2.21·35^-s + 2.17·37^-s − 6.07i·41^-s + 9.21i·43^-s − 3.41·47^-s + ⋯

L(s) = 1

− 0.990i·5^-s − 0.377i·7^-s − 0.185·11^-s − 1.54·13^-s + 0.142i·17^-s + 0.894i·19^-s − 0.177·23^-s + 0.0192·25^-s + 0.880i·29^-s − 0.551i·31^-s − 0.374·35^-s + 0.357·37^-s − 0.949i·41^-s + 1.40i·43^-s − 0.498·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$6048$ = $2^{5} \cdot 3^{3} \cdot 7$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$48.2935$
Root analytic conductor:	$6.94935$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{6048} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.132913368$
$L(\frac12)$	$\approx$	$1.132913368$
$L(\frac{3}{2})$		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 + iT $
good	5	$ 1 + 2.21iT - 5T^{2} $
	11	$ 1 + 0.614T + 11T^{2} $
	13	$ 1 + 5.56T + 13T^{2} $
	17	$ 1 - 0.585iT - 17T^{2} $
	19	$ 1 - 3.89iT - 19T^{2} $
	23	$ 1 + 0.850T + 23T^{2} $
	29	$ 1 - 4.74iT - 29T^{2} $
	31	$ 1 + 3.07iT - 31T^{2} $
	37	$ 1 - 2.17T + 37T^{2} $
	41	$ 1 + 6.07iT - 41T^{2} $
	43	$ 1 - 9.21iT - 43T^{2} $
	47	$ 1 + 3.41T + 47T^{2} $
	53	$ 1 - 4.47iT - 53T^{2} $
	59	$ 1 - 13.9T + 59T^{2} $
	61	$ 1 + 8.92T + 61T^{2} $
	67	$ 1 - 4.83iT - 67T^{2} $
	71	$ 1 + 5.33T + 71T^{2} $
	73	$ 1 + 9.28T + 73T^{2} $
	79	$ 1 + 1.29iT - 79T^{2} $
	83	$ 1 - 14.5T + 83T^{2} $
	89	$ 1 - 11.4iT - 89T^{2} $
	97	$ 1 - 7.27T + 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

−8.037681291870861842998049058148, −7.60449499152718438598184608489, −6.83688173657454595450769553474, −5.92554432001766297641700165650, −5.16794728108004199204999415425, −4.63941010776604061777538875572, −3.88854853685782810234923855870, −2.85499499830057701871566955108, −1.88791899265750238759320607617, −0.874405994031680972316733739956, 0.34256551091459152926923513888, 2.01869548569584547670410123845, 2.67445716043996174366511224728, 3.29995797474232562393786406912, 4.46971878115254365059242741488, 5.05843716992672568202909606141, 5.90694296920907023652283000592, 6.72580832654775086262332114759, 7.19902256249125588929249701667, 7.83589451834374899195925830247

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.132913368\)
\(L(\frac12)\)	\(\approx\)	\(1.132913368\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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