LMFDB - L-function 2-4032-28.27-c1-0-30

Dirichlet series

L(s) = 1

+ 1.72i·5^-s + (2.63 − 0.222i)7^-s + 6.17i·11^-s + 2.82i·13^-s − 1.72i·17^-s + 5.90·19^-s + 1.54i·23^-s + 2.01·25^-s + 8.28·29^-s − 4.62·31^-s + (0.384 + 4.55i)35^-s − 2.24·37^-s + 3.92i·41^-s − 10.1i·43^-s − 4.88·47^-s + ⋯

L(s) = 1

+ 0.772i·5^-s + (0.996 − 0.0839i)7^-s + 1.86i·11^-s + 0.784i·13^-s − 0.419i·17^-s + 1.35·19^-s + 0.322i·23^-s + 0.402·25^-s + 1.53·29^-s − 0.831·31^-s + (0.0649 + 0.770i)35^-s − 0.368·37^-s + 0.613i·41^-s − 1.54i·43^-s − 0.713·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0839 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0839 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$-0.0839 - 0.996i$
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:	$(2,\ 4032,\ (\ :1/2),\ -0.0839 - 0.996i)$

Particular Values

$L(1)$	$\approx$	$2.203603384$
$L(\frac12)$	$\approx$	$2.203603384$
$L(\frac{3}{2})$		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 + (-2.63 + 0.222i)T $
good	5	$ 1 - 1.72iT - 5T^{2} $
	11	$ 1 - 6.17iT - 11T^{2} $
	13	$ 1 - 2.82iT - 13T^{2} $
	17	$ 1 + 1.72iT - 17T^{2} $
	19	$ 1 - 5.90T + 19T^{2} $
	23	$ 1 - 1.54iT - 23T^{2} $
	29	$ 1 - 8.28T + 29T^{2} $
	31	$ 1 + 4.62T + 31T^{2} $
	37	$ 1 + 2.24T + 37T^{2} $
	41	$ 1 - 3.92iT - 41T^{2} $
	43	$ 1 + 10.1iT - 43T^{2} $
	47	$ 1 + 4.88T + 47T^{2} $
	53	$ 1 + 9.17T + 53T^{2} $
	59	$ 1 - 9.65T + 59T^{2} $
	61	$ 1 - 14.2iT - 61T^{2} $
	67	$ 1 + 6.64iT - 67T^{2} $
	71	$ 1 - 5.00iT - 71T^{2} $
	73	$ 1 - 2.19iT - 73T^{2} $
	79	$ 1 + 9.55iT - 79T^{2} $
	83	$ 1 + 12.2T + 83T^{2} $
	89	$ 1 - 9.16iT - 89T^{2} $
	97	$ 1 + 17.1iT - 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

−8.613699081527889528405901088707, −7.70858945357395585438795610730, −7.06684267050639354336908645516, −6.83193796267217045269532473705, −5.51845951635599679354874142345, −4.82574193685717544169487837270, −4.22775804225200556673072086390, −3.10629923907987206345258962080, −2.17547177301721359048092209849, −1.36200658278277885533344074903, 0.69872793542977972406389779136, 1.40452934593673066416288806614, 2.83954013573704678088389114091, 3.52401175744270656387813305564, 4.65470521944727395567656904710, 5.27975347139824332684150079489, 5.81729984435793892884306013874, 6.73932350061421094748403779408, 7.88400201102304483085458807471, 8.242990868979602928061624822867

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.203603384\)
\(L(\frac12)\)	\(\approx\)	\(2.203603384\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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