LMFDB - L-function 2-6048-12.11-c1-0-22

Dirichlet series

L(s) = 1

− 3.14i·5^-s − i·7^-s + 1.41·11^-s + 0.449·13^-s + 2.51i·17^-s + 2.44i·19^-s − 5.65·23^-s − 4.89·25^-s + 8.34i·29^-s + 1.55i·31^-s − 3.14·35^-s + 9·37^-s + 10.0i·41^-s + 8.34i·43^-s − 0.460·47^-s + ⋯

L(s) = 1

− 1.40i·5^-s − 0.377i·7^-s + 0.426·11^-s + 0.124·13^-s + 0.608i·17^-s + 0.561i·19^-s − 1.17·23^-s − 0.979·25^-s + 1.54i·29^-s + 0.278i·31^-s − 0.531·35^-s + 1.47·37^-s + 1.57i·41^-s + 1.27i·43^-s − 0.0672·47^-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

$L(1)$	$\approx$	$1.417032918$
$L(\frac12)$	$\approx$	$1.417032918$
$L(\frac{3}{2})$		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 + 3.14iT - 5T^{2} $
	11	$ 1 - 1.41T + 11T^{2} $
	13	$ 1 - 0.449T + 13T^{2} $
	17	$ 1 - 2.51iT - 17T^{2} $
	19	$ 1 - 2.44iT - 19T^{2} $
	23	$ 1 + 5.65T + 23T^{2} $
	29	$ 1 - 8.34iT - 29T^{2} $
	31	$ 1 - 1.55iT - 31T^{2} $
	37	$ 1 - 9T + 37T^{2} $
	41	$ 1 - 10.0iT - 41T^{2} $
	43	$ 1 - 8.34iT - 43T^{2} $
	47	$ 1 + 0.460T + 47T^{2} $
	53	$ 1 - 5.02iT - 53T^{2} $
	59	$ 1 + 11.4T + 59T^{2} $
	61	$ 1 + 5.55T + 61T^{2} $
	67	$ 1 - 13.7iT - 67T^{2} $
	71	$ 1 - 5.65T + 71T^{2} $
	73	$ 1 - 9.79T + 73T^{2} $
	79	$ 1 - 0.348iT - 79T^{2} $
	83	$ 1 + 2.36T + 83T^{2} $
	89	$ 1 + 1.55iT - 89T^{2} $
	97	$ 1 - 10.2T + 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

−8.022574431226840044227116539195, −7.81805603867569098139805871904, −6.59233943723062745994363894773, −6.04742797466770544209926880633, −5.22740169917542588800975254649, −4.46174661187780724965442124356, −3.99065311733299973274966688552, −2.96356768244291460123816524908, −1.59713539203354763884373306138, −1.10649070392080508803323222944, 0.38409991216973001025554956948, 2.05211161776278149583071426150, 2.57860132009858808037705670405, 3.50648176915530664283888849987, 4.17428664376833229601595568041, 5.18251293407405123340489082103, 6.18871567353129440366402976078, 6.37269793498989187416297795351, 7.35241282308613114854482692925, 7.76906569798388112834364854546

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.417032918\)
\(L(\frac12)\)	\(\approx\)	\(1.417032918\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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