LMFDB - L-function 2-3744-24.11-c1-0-30

Dirichlet series

L(s) = 1

− 2.22·5^-s + 0.534i·7^-s − 2.35i·11^-s − i·13^-s + 4.78i·17^-s + 4.25·19^-s + 2.55·23^-s − 0.0600·25^-s − 8.32·29^-s − 0.284i·31^-s − 1.18i·35^-s − 6.30i·37^-s + 4.33i·41^-s + 0.666·43^-s + 1.10·47^-s + ⋯

L(s) = 1

− 0.993·5^-s + 0.202i·7^-s − 0.710i·11^-s − 0.277i·13^-s + 1.16i·17^-s + 0.975·19^-s + 0.532·23^-s − 0.0120·25^-s − 1.54·29^-s − 0.0510i·31^-s − 0.200i·35^-s − 1.03i·37^-s + 0.676i·41^-s + 0.101·43^-s + 0.160·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3744$ = $2^{5} \cdot 3^{2} \cdot 13$
Sign:	$0.188 + 0.982i$
Analytic conductor:	$29.8959$
Root analytic conductor:	$5.46772$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3744} (2159, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3744,\ (\ :1/2),\ 0.188 + 0.982i)$

Particular Values

$L(1)$	$\approx$	$1.033451544$
$L(\frac12)$	$\approx$	$1.033451544$
$L(\frac{3}{2})$		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 $
	13	$ 1 + iT $
good	5	$ 1 + 2.22T + 5T^{2} $
	7	$ 1 - 0.534iT - 7T^{2} $
	11	$ 1 + 2.35iT - 11T^{2} $
	17	$ 1 - 4.78iT - 17T^{2} $
	19	$ 1 - 4.25T + 19T^{2} $
	23	$ 1 - 2.55T + 23T^{2} $
	29	$ 1 + 8.32T + 29T^{2} $
	31	$ 1 + 0.284iT - 31T^{2} $
	37	$ 1 + 6.30iT - 37T^{2} $
	41	$ 1 - 4.33iT - 41T^{2} $
	43	$ 1 - 0.666T + 43T^{2} $
	47	$ 1 - 1.10T + 47T^{2} $
	53	$ 1 + 1.42T + 53T^{2} $
	59	$ 1 + 3.21iT - 59T^{2} $
	61	$ 1 - 4.98iT - 61T^{2} $
	67	$ 1 + 0.658T + 67T^{2} $
	71	$ 1 + 12.3T + 71T^{2} $
	73	$ 1 - 14.1T + 73T^{2} $
	79	$ 1 + 11.8iT - 79T^{2} $
	83	$ 1 + 14.7iT - 83T^{2} $
	89	$ 1 - 1.91iT - 89T^{2} $
	97	$ 1 + 7.69T + 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

−8.270415192431888218842754979809, −7.66444234523587199766108614081, −7.07818484308067553949265976505, −5.94068907088403734564138059679, −5.51726384066065833938760017573, −4.37460838297962152435364064713, −3.66877463139885942654046231812, −3.00324522419839074319712631215, −1.68528864018252893555958484775, −0.37617137899682389364061662356, 0.959187271201361667926670698411, 2.27683798393100452044886359280, 3.32701464028071667796881779485, 4.02967775110571582544559394301, 4.86175879195814680230162633962, 5.52571031986464230914063515906, 6.71297312586865604281589056249, 7.37316481319094681521153956966, 7.67585223858002155356818657023, 8.669875316027393312108013856848

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.033451544\)
\(L(\frac12)\)	\(\approx\)	\(1.033451544\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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