LMFDB - L-function 2-4032-21.20-c1-0-20

Dirichlet series

L(s) = 1

− 1.08·5^-s + (−2.10 − 1.60i)7^-s − 1.23i·11^-s + 3.69i·13^-s − 6.30·17^-s − 7.76i·19^-s + 7.17i·23^-s − 3.82·25^-s + 1.41i·29^-s + 4.54i·31^-s + (2.27 + 1.74i)35^-s − 2.82·37^-s + 9.37·41^-s + 7.68·43^-s + 10.9·47^-s + ⋯

L(s) = 1

− 0.484·5^-s + (−0.794 − 0.607i)7^-s − 0.371i·11^-s + 1.02i·13^-s − 1.53·17^-s − 1.78i·19^-s + 1.49i·23^-s − 0.765·25^-s + 0.262i·29^-s + 0.816i·31^-s + (0.384 + 0.294i)35^-s − 0.464·37^-s + 1.46·41^-s + 1.17·43^-s + 1.60·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$0.954 - 0.297i$
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:	$(2,\ 4032,\ (\ :1/2),\ 0.954 - 0.297i)$

Particular Values

$L(1)$	$\approx$	$1.060965114$
$L(\frac12)$	$\approx$	$1.060965114$
$L(\frac{3}{2})$		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 + (2.10 + 1.60i)T $
good	5	$ 1 + 1.08T + 5T^{2} $
	11	$ 1 + 1.23iT - 11T^{2} $
	13	$ 1 - 3.69iT - 13T^{2} $
	17	$ 1 + 6.30T + 17T^{2} $
	19	$ 1 + 7.76iT - 19T^{2} $
	23	$ 1 - 7.17iT - 23T^{2} $
	29	$ 1 - 1.41iT - 29T^{2} $
	31	$ 1 - 4.54iT - 31T^{2} $
	37	$ 1 + 2.82T + 37T^{2} $
	41	$ 1 - 9.37T + 41T^{2} $
	43	$ 1 - 7.68T + 43T^{2} $
	47	$ 1 - 10.9T + 47T^{2} $
	53	$ 1 + 5.41iT - 53T^{2} $
	59	$ 1 + 6.43T + 59T^{2} $
	61	$ 1 + 9.55iT - 61T^{2} $
	67	$ 1 - 14.3T + 67T^{2} $
	71	$ 1 - 10.6iT - 71T^{2} $
	73	$ 1 + 4.59iT - 73T^{2} $
	79	$ 1 + 9.42T + 79T^{2} $
	83	$ 1 + 1.88T + 83T^{2} $
	89	$ 1 + 5.04T + 89T^{2} $
	97	$ 1 + 5.86iT - 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

−8.645133552533689846624074666941, −7.51930140292655862335730400230, −7.03415255566785219333501234328, −6.48142771718493769925072473226, −5.53285898580179516969449831210, −4.47152060262589015670945260927, −3.99453389477135800055275602960, −3.07029924721182815273987490676, −2.08800306983987690405415543787, −0.65987932129543996026152924071, 0.48455761563029191750948343145, 2.14072186309762356823910429857, 2.80642176236956661924479706469, 3.94850795805984956822055182553, 4.38715502879830520158117405003, 5.77557472720273776906124894323, 5.96754843140118953687736651433, 6.97576675919501487160351646461, 7.74344233724581754809291656078, 8.374733094770543295383881070232

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.060965114\)
\(L(\frac12)\)	\(\approx\)	\(1.060965114\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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