LMFDB - L-function 2-84e2-21.20-c1-0-67

Dirichlet series

L(s) = 1

+ 1.84·5^-s + 4i·11^-s − 6.62i·13^-s + 2.29·17^-s − 3.06i·19^-s + 5.65i·23^-s − 1.58·25^-s − 9.65i·29^-s − 10.4i·31^-s − 0.242·37^-s − 0.765·41^-s − 9.65·43^-s + 3.06·47^-s + 0.242i·53^-s + 7.39i·55^-s + ⋯

L(s) = 1

+ 0.826·5^-s + 1.20i·11^-s − 1.83i·13^-s + 0.556·17^-s − 0.702i·19^-s + 1.17i·23^-s − 0.317·25^-s − 1.79i·29^-s − 1.87i·31^-s − 0.0398·37^-s − 0.119·41^-s − 1.47·43^-s + 0.446·47^-s + 0.0333i·53^-s + 0.996i·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$-0.192 + 0.981i$
Analytic conductor:	$56.3424$
Root analytic conductor:	$7.50616$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7056} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7056,\ (\ :1/2),\ -0.192 + 0.981i)$

Particular Values

$L(1)$	$\approx$	$1.674862687$
$L(\frac12)$	$\approx$	$1.674862687$
$L(\frac{3}{2})$		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 $
good	5	$ 1 - 1.84T + 5T^{2} $
	11	$ 1 - 4iT - 11T^{2} $
	13	$ 1 + 6.62iT - 13T^{2} $
	17	$ 1 - 2.29T + 17T^{2} $
	19	$ 1 + 3.06iT - 19T^{2} $
	23	$ 1 - 5.65iT - 23T^{2} $
	29	$ 1 + 9.65iT - 29T^{2} $
	31	$ 1 + 10.4iT - 31T^{2} $
	37	$ 1 + 0.242T + 37T^{2} $
	41	$ 1 + 0.765T + 41T^{2} $
	43	$ 1 + 9.65T + 43T^{2} $
	47	$ 1 - 3.06T + 47T^{2} $
	53	$ 1 - 0.242iT - 53T^{2} $
	59	$ 1 + 10.4T + 59T^{2} $
	61	$ 1 - 7.97iT - 61T^{2} $
	67	$ 1 + 4T + 67T^{2} $
	71	$ 1 + 2.34iT - 71T^{2} $
	73	$ 1 + 9.23iT - 73T^{2} $
	79	$ 1 - 5.65T + 79T^{2} $
	83	$ 1 + 11.7T + 83T^{2} $
	89	$ 1 + 3.11T + 89T^{2} $
	97	$ 1 - 9.23iT - 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

−7.68678474289255577028225575026, −7.23578105200513804695932755964, −6.05714342227167289763407220063, −5.78655138252385624527631124243, −5.00017226436541422087218581063, −4.20544476708905407587094764531, −3.21574523828156553887537486494, −2.42688157715094706705965024853, −1.63102494644869837298452209940, −0.38489257382608950158206945607, 1.28639889275626464758338100739, 1.89884083342095286427616151569, 3.03311864832550748657775030639, 3.67611708897846973769572212280, 4.69538010009099424386957633780, 5.32740046495476720732004371531, 6.16418296045621049650559422877, 6.60726305773789577538836344052, 7.28214167324811021577020626928, 8.437031670964832249984934976242

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.674862687\)
\(L(\frac12)\)	\(\approx\)	\(1.674862687\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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