LMFDB - L-function 2-3332-68.67-c0-0-11

Dirichlet series

L(s) = 1

− 2^-s + 1.73·3^-s + 4^-s − 1.73·6^-s − 8^-s + 1.99·9^-s − 1.73·11^-s + 1.73·12^-s + 13^-s + 16^-s + 17^-s − 1.99·18^-s + 1.73·22^-s − 1.73·24^-s + 25^-s − 26^-s + 1.73·27^-s − 32^-s − 2.99·33^-s − 34^-s + 1.99·36^-s + 1.73·39^-s − 1.73·44^-s + 1.73·48^-s − 50^-s + 1.73·51^-s + 52^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3332$ = $2^{2} \cdot 7^{2} \cdot 17$
Sign:	$1$
Analytic conductor:	$1.66288$
Root analytic conductor:	$1.28952$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3332} (883, \cdot )$
Primitive:	yes
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(2,\ 3332,\ (\ :0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.526781629$
$L(\frac12)$	$\approx$	$1.526781629$
$L(1)$		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 + T $
	7	$ 1 $
	17	$ 1 - T $
good	3	$ 1 - 1.73T + T^{2} $
	5	$ 1 - T^{2} $
	11	$ 1 + 1.73T + T^{2} $
	13	$ 1 - T + T^{2} $
	19	$ 1 - T^{2} $
	23	$ 1 + T^{2} $
	29	$ 1 - T^{2} $
	31	$ 1 + T^{2} $
	37	$ 1 - T^{2} $
	41	$ 1 - T^{2} $
	43	$ 1 - T^{2} $
	47	$ 1 - T^{2} $
	53	$ 1 + T + T^{2} $
	59	$ 1 - T^{2} $
	61	$ 1 - T^{2} $
	67	$ 1 - T^{2} $
	71	$ 1 - 1.73T + T^{2} $
	73	$ 1 - T^{2} $
	79	$ 1 + 1.73T + T^{2} $
	83	$ 1 - T^{2} $
	89	$ 1 - T + T^{2} $
	97	$ 1 - T^{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

−8.620933075111954257645631536986, −8.200686561090330194341197503467, −7.68363495275501182808579406818, −7.03636006340168503341862007103, −5.98925987369782933819195683307, −4.98726622899125756363525248303, −3.64031054439535941364333605260, −3.00803724595188655270512892113, −2.34786515678965719531469512308, −1.28017951305091991985577267014, 1.28017951305091991985577267014, 2.34786515678965719531469512308, 3.00803724595188655270512892113, 3.64031054439535941364333605260, 4.98726622899125756363525248303, 5.98925987369782933819195683307, 7.03636006340168503341862007103, 7.68363495275501182808579406818, 8.200686561090330194341197503467, 8.620933075111954257645631536986

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 $\F_{q}$

\(L(\frac{1}{2})\)	\(\approx\)	\(1.526781629\)
\(L(\frac12)\)	\(\approx\)	\(1.526781629\)
\(L(1)\)		not available
\(L(1)\)		not available

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