LMFDB - L-function 2-6048-12.11-c1-0-54

Dirichlet series

L(s) = 1

− 1.19i·5^-s + i·7^-s + 1.62·11^-s − 2.09·13^-s + 0.585i·17^-s + 5.89i·19^-s − 7.09·23^-s + 3.56·25^-s − 7.08i·29^-s − 6.72i·31^-s + 1.19·35^-s + 2.17·37^-s − 0.164i·41^-s − 5.75i·43^-s + 3.41·47^-s + ⋯

L(s) = 1

− 0.536i·5^-s + 0.377i·7^-s + 0.491·11^-s − 0.581·13^-s + 0.142i·17^-s + 1.35i·19^-s − 1.47·23^-s + 0.712·25^-s − 1.31i·29^-s − 1.20i·31^-s + 0.202·35^-s + 0.357·37^-s − 0.0256i·41^-s − 0.877i·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

$L(1)$	$\approx$	$1.710053417$
$L(\frac12)$	$\approx$	$1.710053417$
$L(\frac{3}{2})$		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 + 1.19iT - 5T^{2} $
	11	$ 1 - 1.62T + 11T^{2} $
	13	$ 1 + 2.09T + 13T^{2} $
	17	$ 1 - 0.585iT - 17T^{2} $
	19	$ 1 - 5.89iT - 19T^{2} $
	23	$ 1 + 7.09T + 23T^{2} $
	29	$ 1 + 7.08iT - 29T^{2} $
	31	$ 1 + 6.72iT - 31T^{2} $
	37	$ 1 - 2.17T + 37T^{2} $
	41	$ 1 + 0.164iT - 41T^{2} $
	43	$ 1 + 5.75iT - 43T^{2} $
	47	$ 1 - 3.41T + 47T^{2} $
	53	$ 1 + 3.29iT - 53T^{2} $
	59	$ 1 + 2.08T + 59T^{2} $
	61	$ 1 - 4.92T + 61T^{2} $
	67	$ 1 - 12.4iT - 67T^{2} $
	71	$ 1 - 11.2T + 71T^{2} $
	73	$ 1 - 3.97T + 73T^{2} $
	79	$ 1 - 0.702iT - 79T^{2} $
	83	$ 1 - 16.8T + 83T^{2} $
	89	$ 1 + 6.33iT - 89T^{2} $
	97	$ 1 + 6.58T + 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

−8.080804037252855033500940072969, −7.41449131693680332616702229621, −6.40821147203551892466190969270, −5.87215414627648563819356214326, −5.18982031612449762825737517709, −4.20047067404060180860869612956, −3.77995785744651453718868694559, −2.48998799819155797493753611556, −1.80791014689722698321022390770, −0.55284156046337761778354619187, 0.849805310832086401744539908103, 2.04233273438328819867364554648, 2.94021879970568770922825661705, 3.65098217924772739535124201199, 4.63015673294278485125682146676, 5.13963987793000314383938861649, 6.26128563100539264487715267649, 6.77364953585480308868443836102, 7.33523217857380590964340523604, 8.085537773383525637565899360019

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.710053417\)
\(L(\frac12)\)	\(\approx\)	\(1.710053417\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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