LMFDB - L-function 2-4032-12.11-c1-0-27

Dirichlet series

L(s) = 1

− 2.20i·5^-s + i·7^-s − 0.794·11^-s + 5.12·13^-s − 0.620i·17^-s − 4i·19^-s + 0.794·23^-s + 0.123·25^-s + 3.00i·29^-s + 6.24i·31^-s + 2.20·35^-s + 11.1·37^-s − 3.44i·41^-s − 4i·43^-s − 12.9·47^-s + ⋯

L(s) = 1

− 0.987i·5^-s + 0.377i·7^-s − 0.239·11^-s + 1.42·13^-s − 0.150i·17^-s − 0.917i·19^-s + 0.165·23^-s + 0.0246·25^-s + 0.557i·29^-s + 1.12i·31^-s + 0.373·35^-s + 1.82·37^-s − 0.538i·41^-s − 0.609i·43^-s − 1.88·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$0.577 + 0.816i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (575, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ 0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$1.984789884$
$L(\frac12)$	$\approx$	$1.984789884$
$L(\frac{3}{2})$		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 - iT $
good	5	$ 1 + 2.20iT - 5T^{2} $
	11	$ 1 + 0.794T + 11T^{2} $
	13	$ 1 - 5.12T + 13T^{2} $
	17	$ 1 + 0.620iT - 17T^{2} $
	19	$ 1 + 4iT - 19T^{2} $
	23	$ 1 - 0.794T + 23T^{2} $
	29	$ 1 - 3.00iT - 29T^{2} $
	31	$ 1 - 6.24iT - 31T^{2} $
	37	$ 1 - 11.1T + 37T^{2} $
	41	$ 1 + 3.44iT - 41T^{2} $
	43	$ 1 + 4iT - 43T^{2} $
	47	$ 1 + 12.9T + 47T^{2} $
	53	$ 1 - 3.00iT - 53T^{2} $
	59	$ 1 + 1.58T + 59T^{2} $
	61	$ 1 - 2T + 61T^{2} $
	67	$ 1 + 5.12iT - 67T^{2} $
	71	$ 1 - 8.03T + 71T^{2} $
	73	$ 1 - 1.12T + 73T^{2} $
	79	$ 1 + 11.3iT - 79T^{2} $
	83	$ 1 - 14.4T + 83T^{2} $
	89	$ 1 - 2.20iT - 89T^{2} $
	97	$ 1 - 1.12T + 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

−8.457217522292583676345440006018, −7.78466624466667835662985901786, −6.75188849087867554635704721062, −6.12633153401097284299040270990, −5.18481135765589686726674849957, −4.75525235631373661430450016691, −3.71837068897237738702017402258, −2.85332183506924646546788812807, −1.65055234328606156615737896406, −0.70914409020238024244004452469, 1.00897253779843828827818059456, 2.19712465555702103517502153411, 3.19881341621709906031824415065, 3.82028946789270103474143499287, 4.68558312673307324548018693218, 5.91467307406879652276158370697, 6.25889658774309227486509427068, 7.02878775825088321031729758044, 8.004012667076370785654502301942, 8.214646458123272358517523407195

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.984789884\)
\(L(\frac12)\)	\(\approx\)	\(1.984789884\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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