LMFDB - L-function 2-4608-8.5-c1-0-10

Dirichlet series

L(s) = 1

− 3.41i·5^-s + 0.585·7^-s + 2i·11^-s + 2.82i·13^-s + 7.65·17^-s + 5.65i·19^-s − 6.82·23^-s − 6.65·25^-s + 3.41i·29^-s − 7.41·31^-s − 2i·35^-s + 1.65i·37^-s − 0.343·41^-s + 9.65i·43^-s − 4.48·47^-s + ⋯

L(s) = 1

− 1.52i·5^-s + 0.221·7^-s + 0.603i·11^-s + 0.784i·13^-s + 1.85·17^-s + 1.29i·19^-s − 1.42·23^-s − 1.33·25^-s + 0.634i·29^-s − 1.33·31^-s − 0.338i·35^-s + 0.272i·37^-s − 0.0535·41^-s + 1.47i·43^-s − 0.654·47^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$4608$ = $2^{9} \cdot 3^{2}$
Sign:	$-i$
Analytic conductor:	$36.7950$
Root analytic conductor:	$6.06589$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4608} (2305, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4608,\ (\ :1/2),\ -i)$

Particular Values

$L(1)$	$\approx$	$1.040261811$
$L(\frac12)$	$\approx$	$1.040261811$
$L(\frac{3}{2})$		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 + 3.41iT - 5T^{2} $
	7	$ 1 - 0.585T + 7T^{2} $
	11	$ 1 - 2iT - 11T^{2} $
	13	$ 1 - 2.82iT - 13T^{2} $
	17	$ 1 - 7.65T + 17T^{2} $
	19	$ 1 - 5.65iT - 19T^{2} $
	23	$ 1 + 6.82T + 23T^{2} $
	29	$ 1 - 3.41iT - 29T^{2} $
	31	$ 1 + 7.41T + 31T^{2} $
	37	$ 1 - 1.65iT - 37T^{2} $
	41	$ 1 + 0.343T + 41T^{2} $
	43	$ 1 - 9.65iT - 43T^{2} $
	47	$ 1 + 4.48T + 47T^{2} $
	53	$ 1 - 7.89iT - 53T^{2} $
	59	$ 1 + 4iT - 59T^{2} $
	61	$ 1 + 1.65iT - 61T^{2} $
	67	$ 1 + 8iT - 67T^{2} $
	71	$ 1 + 14.8T + 71T^{2} $
	73	$ 1 + 9.65T + 73T^{2} $
	79	$ 1 + 14.2T + 79T^{2} $
	83	$ 1 - 13.3iT - 83T^{2} $
	89	$ 1 - 2T + 89T^{2} $
	97	$ 1 + 9.31T + 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

−8.336746563082434048668877742592, −7.957183912395862851880549298396, −7.26054574348685493102133065803, −6.05789979171117677786025044866, −5.57031318318754740468010663955, −4.75387054156745692400070625279, −4.14907299330438271386954608392, −3.28572263298868847782614675995, −1.70647320274923551760404857302, −1.38929582007157612389015645166, 0.27711565706114710016875860092, 1.80003910916172094604559066927, 2.90882498536768087829469425631, 3.30481885441278746030408684627, 4.21729156958271808481788530140, 5.58396203307732431183139088584, 5.75968445501680270211203448855, 6.81431002346493054622580990997, 7.40059968273424425885742643519, 7.968286522095441317591790827302

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.040261811\)
\(L(\frac12)\)	\(\approx\)	\(1.040261811\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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