LMFDB - L-function 2-4032-28.27-c1-0-24

Dirichlet series

L(s) = 1

− 1.69i·5^-s + (2.13 + 1.56i)7^-s − 0.794i·11^-s + 1.87i·13^-s + 4.34i·17^-s + 2.39·19^-s + 3.62i·23^-s + 2.12·25^-s − 4.41·29^-s + 1.87·31^-s + (2.64 − 3.62i)35^-s + 3.12·37^-s + 4.34i·41^-s − 0.876i·43^-s − 12.0·47^-s + ⋯

L(s) = 1

− 0.758i·5^-s + (0.807 + 0.590i)7^-s − 0.239i·11^-s + 0.519i·13^-s + 1.05i·17^-s + 0.550·19^-s + 0.755i·23^-s + 0.424·25^-s − 0.820·29^-s + 0.336·31^-s + (0.447 − 0.612i)35^-s + 0.513·37^-s + 0.678i·41^-s − 0.133i·43^-s − 1.76·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$0.590 - 0.807i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (3583, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ 0.590 - 0.807i)$

Particular Values

$L(1)$	$\approx$	$1.912880124$
$L(\frac12)$	$\approx$	$1.912880124$
$L(\frac{3}{2})$		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 + (-2.13 - 1.56i)T $
good	5	$ 1 + 1.69iT - 5T^{2} $
	11	$ 1 + 0.794iT - 11T^{2} $
	13	$ 1 - 1.87iT - 13T^{2} $
	17	$ 1 - 4.34iT - 17T^{2} $
	19	$ 1 - 2.39T + 19T^{2} $
	23	$ 1 - 3.62iT - 23T^{2} $
	29	$ 1 + 4.41T + 29T^{2} $
	31	$ 1 - 1.87T + 31T^{2} $
	37	$ 1 - 3.12T + 37T^{2} $
	41	$ 1 - 4.34iT - 41T^{2} $
	43	$ 1 + 0.876iT - 43T^{2} $
	47	$ 1 + 12.0T + 47T^{2} $
	53	$ 1 + 10.0T + 53T^{2} $
	59	$ 1 - 6.78T + 59T^{2} $
	61	$ 1 - 11.4iT - 61T^{2} $
	67	$ 1 - 0.876iT - 67T^{2} $
	71	$ 1 - 5.21iT - 71T^{2} $
	73	$ 1 - 3.74iT - 73T^{2} $
	79	$ 1 + 3.12iT - 79T^{2} $
	83	$ 1 - 12.0T + 83T^{2} $
	89	$ 1 - 7.73iT - 89T^{2} $
	97	$ 1 - 13.3iT - 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

−8.387411866584069556420465778836, −8.107638569537243687791831393314, −7.16501176808204126596327688764, −6.20439193712601895253370330914, −5.51153878639675920040915609952, −4.85143111008595718884719451181, −4.10725200745277372024186429432, −3.09921520412600746275306307270, −1.90990515573070604248286020233, −1.19250132756309188254851350512, 0.58981595801362345489502357644, 1.85477439890158649281489842176, 2.86859457011213836664704503569, 3.60613165451519693590542799130, 4.71368729055212306882658441068, 5.15014457906568784465824928780, 6.27748373473460908278601032403, 6.93958271284475916348759699139, 7.62100138415434269718532855830, 8.108091707036370354027328664970

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.912880124\)
\(L(\frac12)\)	\(\approx\)	\(1.912880124\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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