LMFDB - L-function 2-4608-12.11-c1-0-13

Dirichlet series

L(s) = 1

− 4.24i·7^-s + 4·11^-s − 6·13^-s + 4.24i·17^-s + 2.82i·19^-s − 6·23^-s + 5·25^-s + 8.48i·29^-s + 4.24i·31^-s − 6·37^-s − 1.41i·41^-s + 2.82i·43^-s + 6·47^-s − 10.9·49^-s + 8.48i·53^-s + ⋯

L(s) = 1

− 1.60i·7^-s + 1.20·11^-s − 1.66·13^-s + 1.02i·17^-s + 0.648i·19^-s − 1.25·23^-s + 25^-s + 1.57i·29^-s + 0.762i·31^-s − 0.986·37^-s − 0.220i·41^-s + 0.431i·43^-s + 0.875·47^-s − 1.57·49^-s + 1.16i·53^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4608$ = $2^{9} \cdot 3^{2}$
Sign:	$0.577 - 0.816i$
Analytic conductor:	$36.7950$
Root analytic conductor:	$6.06589$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4608} (4607, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4608,\ (\ :1/2),\ 0.577 - 0.816i)$

Particular Values

$L(1)$	$\approx$	$1.383117932$
$L(\frac12)$	$\approx$	$1.383117932$
$L(\frac{3}{2})$		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 $
good	5	$ 1 - 5T^{2} $
	7	$ 1 + 4.24iT - 7T^{2} $
	11	$ 1 - 4T + 11T^{2} $
	13	$ 1 + 6T + 13T^{2} $
	17	$ 1 - 4.24iT - 17T^{2} $
	19	$ 1 - 2.82iT - 19T^{2} $
	23	$ 1 + 6T + 23T^{2} $
	29	$ 1 - 8.48iT - 29T^{2} $
	31	$ 1 - 4.24iT - 31T^{2} $
	37	$ 1 + 6T + 37T^{2} $
	41	$ 1 + 1.41iT - 41T^{2} $
	43	$ 1 - 2.82iT - 43T^{2} $
	47	$ 1 - 6T + 47T^{2} $
	53	$ 1 - 8.48iT - 53T^{2} $
	59	$ 1 + 4T + 59T^{2} $
	61	$ 1 - 6T + 61T^{2} $
	67	$ 1 - 11.3iT - 67T^{2} $
	71	$ 1 - 6T + 71T^{2} $
	73	$ 1 - 6T + 73T^{2} $
	79	$ 1 - 4.24iT - 79T^{2} $
	83	$ 1 - 16T + 83T^{2} $
	89	$ 1 + 12.7iT - 89T^{2} $
	97	$ 1 + 12T + 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

−8.379242369129545391056555770967, −7.54836601528391915874341346443, −7.00541818793748716783120293566, −6.50033308898321330160115165509, −5.43456788876945991080903544483, −4.53845935531958948573654704694, −3.95998620672111673457320435944, −3.24675181572872360970706542169, −1.88672077314071183445224226437, −1.03182485936910985551663084346, 0.42060422220265417722496646386, 2.15372077953322848092943999975, 2.45500455513515700810367104301, 3.59157795495890434429487960703, 4.67740197380450760806122119072, 5.18256315039191513160615503854, 6.05531711658694102965216739370, 6.69741620459850175635473403581, 7.47492027985152493306872111186, 8.281727875996340205926728840513

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.383117932\)
\(L(\frac12)\)	\(\approx\)	\(1.383117932\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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