LMFDB - L-function 2-4032-56.27-c1-0-14

Dirichlet series

L(s) = 1

+ 1.11·5^-s + (−1.37 + 2.26i)7^-s − 0.812·11^-s + 2·13^-s + 3.55i·17^-s − 4.52i·19^-s − 3.63i·23^-s − 3.75·25^-s + 8.95i·29^-s − 5.08·31^-s + (−1.53 + 2.52i)35^-s + 0.718i·37^-s + 10.4i·41^-s + 1.11·43^-s + 8.55·47^-s + ⋯

L(s) = 1

+ 0.498·5^-s + (−0.518 + 0.854i)7^-s − 0.245·11^-s + 0.554·13^-s + 0.862i·17^-s − 1.03i·19^-s − 0.758i·23^-s − 0.751·25^-s + 1.66i·29^-s − 0.914·31^-s + (−0.258 + 0.426i)35^-s + 0.118i·37^-s + 1.63i·41^-s + 0.170·43^-s + 1.24·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$-0.691 - 0.722i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (1567, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ -0.691 - 0.722i)$

Particular Values

$L(1)$	$\approx$	$1.056216052$
$L(\frac12)$	$\approx$	$1.056216052$
$L(\frac{3}{2})$		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 + (1.37 - 2.26i)T $
good	5	$ 1 - 1.11T + 5T^{2} $
	11	$ 1 + 0.812T + 11T^{2} $
	13	$ 1 - 2T + 13T^{2} $
	17	$ 1 - 3.55iT - 17T^{2} $
	19	$ 1 + 4.52iT - 19T^{2} $
	23	$ 1 + 3.63iT - 23T^{2} $
	29	$ 1 - 8.95iT - 29T^{2} $
	31	$ 1 + 5.08T + 31T^{2} $
	37	$ 1 - 0.718iT - 37T^{2} $
	41	$ 1 - 10.4iT - 41T^{2} $
	43	$ 1 - 1.11T + 43T^{2} $
	47	$ 1 - 8.55T + 47T^{2} $
	53	$ 1 - 5.08iT - 53T^{2} $
	59	$ 1 - 2.23iT - 59T^{2} $
	61	$ 1 - 5.27T + 61T^{2} $
	67	$ 1 + 15.1T + 67T^{2} $
	71	$ 1 + 5.87iT - 71T^{2} $
	73	$ 1 - 3.86iT - 73T^{2} $
	79	$ 1 - 1.70iT - 79T^{2} $
	83	$ 1 - 7.27iT - 83T^{2} $
	89	$ 1 + 3.37iT - 89T^{2} $
	97	$ 1 + 8.55iT - 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

−8.877780034240087137927480830058, −8.114326530975171893694304586662, −7.14747554667080398991847376894, −6.39174352521593641358052342842, −5.82141408435978332016306628593, −5.13069543245349241813532731035, −4.14447033083242216963169350957, −3.14687698555668675139439737873, −2.41236765596003841162529901204, −1.37576482807414501065882195864, 0.29376694343137726575926845960, 1.56042300046404868427581920327, 2.57281981010476294184996595781, 3.69801660414401146521958575335, 4.12121611692954242805494930970, 5.41613643506916836045933398709, 5.85744783767649351633033221142, 6.72026272445872391627327363303, 7.50142518298431981738469837582, 7.978612439536502563548658395715

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.056216052\)
\(L(\frac12)\)	\(\approx\)	\(1.056216052\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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