LMFDB - L-function 2-4032-21.20-c1-0-11

Dirichlet series

L(s) = 1

+ 0.841·5^-s + (1.16 − 2.37i)7^-s + 3.64i·11^-s + 5.53i·13^-s − 0.841·17^-s + 3.06i·19^-s + 3.64i·23^-s − 4.29·25^-s − 8.89i·29^-s + 7.82i·31^-s + (0.979 − 1.99i)35^-s + 3.29·37^-s − 8.66·41^-s + 2.32·43^-s − 9.10·47^-s + ⋯

L(s) = 1

+ 0.376·5^-s + (0.439 − 0.898i)7^-s + 1.09i·11^-s + 1.53i·13^-s − 0.204·17^-s + 0.704i·19^-s + 0.760i·23^-s − 0.858·25^-s − 1.65i·29^-s + 1.40i·31^-s + (0.165 − 0.338i)35^-s + 0.541·37^-s − 1.35·41^-s + 0.354·43^-s − 1.32·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$-0.479 - 0.877i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (3905, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ -0.479 - 0.877i)$

Particular Values

$L(1)$	$\approx$	$1.246595395$
$L(\frac12)$	$\approx$	$1.246595395$
$L(\frac{3}{2})$		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 + (-1.16 + 2.37i)T $
good	5	$ 1 - 0.841T + 5T^{2} $
	11	$ 1 - 3.64iT - 11T^{2} $
	13	$ 1 - 5.53iT - 13T^{2} $
	17	$ 1 + 0.841T + 17T^{2} $
	19	$ 1 - 3.06iT - 19T^{2} $
	23	$ 1 - 3.64iT - 23T^{2} $
	29	$ 1 + 8.89iT - 29T^{2} $
	31	$ 1 - 7.82iT - 31T^{2} $
	37	$ 1 - 3.29T + 37T^{2} $
	41	$ 1 + 8.66T + 41T^{2} $
	43	$ 1 - 2.32T + 43T^{2} $
	47	$ 1 + 9.10T + 47T^{2} $
	53	$ 1 - 4.24iT - 53T^{2} $
	59	$ 1 + 11.0T + 59T^{2} $
	61	$ 1 - 9.10iT - 61T^{2} $
	67	$ 1 + 8.48T + 67T^{2} $
	71	$ 1 + 0.354iT - 71T^{2} $
	73	$ 1 + 14.6iT - 73T^{2} $
	79	$ 1 + 8.48T + 79T^{2} $
	83	$ 1 + 9.10T + 83T^{2} $
	89	$ 1 + 6.97T + 89T^{2} $
	97	$ 1 + 3.57iT - 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

−8.708893094501864231391892159979, −7.78390888566696694679967604601, −7.27669412429608536353376566197, −6.55265204520783804677060191544, −5.81293828366993569415798242989, −4.61308108232383352456161213805, −4.38427268416209951777545505923, −3.36382756879357226890751171784, −1.95650334804613924663496306288, −1.53749020976416491014013829028, 0.33164336414371821542641563367, 1.66421148264558399284246820058, 2.75145679444914412800801595387, 3.29256213403328471333077291361, 4.56814831305386717357170830414, 5.38953253606081149195104610411, 5.83868876607888242066709641102, 6.58076325650646813850208391440, 7.64191670926827733685469017771, 8.341044542568816612378776914490

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.246595395\)
\(L(\frac12)\)	\(\approx\)	\(1.246595395\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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