LMFDB - L-function 2-7524-209.208-c1-0-57

Dirichlet series

L(s) = 1

− 0.783·5^-s + 3.52i·7^-s + (3.16 + 0.975i)11^-s − 5.92·13^-s − 5.26i·17^-s + (3.88 + 1.97i)19^-s − 6.49·23^-s − 4.38·25^-s + 5.64·29^-s − 10.5i·31^-s − 2.75i·35^-s + 5.33i·37^-s + 5.07·41^-s − 0.0929i·43^-s − 5.65·47^-s + ⋯

L(s) = 1

− 0.350·5^-s + 1.33i·7^-s + (0.955 + 0.294i)11^-s − 1.64·13^-s − 1.27i·17^-s + (0.891 + 0.453i)19^-s − 1.35·23^-s − 0.877·25^-s + 1.04·29^-s − 1.89i·31^-s − 0.466i·35^-s + 0.876i·37^-s + 0.793·41^-s − 0.0141i·43^-s − 0.824·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7524 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7524$ = $2^{2} \cdot 3^{2} \cdot 11 \cdot 19$
Sign:	$0.718 + 0.695i$
Analytic conductor:	$60.0794$
Root analytic conductor:	$7.75109$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7524} (2089, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7524,\ (\ :1/2),\ 0.718 + 0.695i)$

Particular Values

$L(1)$	$\approx$	$1.237146265$
$L(\frac12)$	$\approx$	$1.237146265$
$L(\frac{3}{2})$		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 $
	11	$ 1 + (-3.16 - 0.975i)T $
	19	$ 1 + (-3.88 - 1.97i)T $
good	5	$ 1 + 0.783T + 5T^{2} $
	7	$ 1 - 3.52iT - 7T^{2} $
	13	$ 1 + 5.92T + 13T^{2} $
	17	$ 1 + 5.26iT - 17T^{2} $
	23	$ 1 + 6.49T + 23T^{2} $
	29	$ 1 - 5.64T + 29T^{2} $
	31	$ 1 + 10.5iT - 31T^{2} $
	37	$ 1 - 5.33iT - 37T^{2} $
	41	$ 1 - 5.07T + 41T^{2} $
	43	$ 1 + 0.0929iT - 43T^{2} $
	47	$ 1 + 5.65T + 47T^{2} $
	53	$ 1 + 3.79iT - 53T^{2} $
	59	$ 1 + 5.90iT - 59T^{2} $
	61	$ 1 - 2.59iT - 61T^{2} $
	67	$ 1 - 9.38iT - 67T^{2} $
	71	$ 1 + 9.09iT - 71T^{2} $
	73	$ 1 + 12.8iT - 73T^{2} $
	79	$ 1 + 0.843T + 79T^{2} $
	83	$ 1 + 12.3iT - 83T^{2} $
	89	$ 1 - 2.44iT - 89T^{2} $
	97	$ 1 - 5.08iT - 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

−7.73053636508143988889495000467, −7.27446817602925942867983681469, −6.29166933284677411039224664270, −5.77629142698218228056813955035, −4.90972070114469754892317596869, −4.38594278182533582819452398333, −3.34181728563091396833500481785, −2.51001706477601204200507785715, −1.89179401737457383401587197561, −0.37121495960881776968253007596, 0.826315398321213327094385406442, 1.76988388509446157154194571603, 2.92047901588741692431607262681, 3.85216764162134876368225179245, 4.21512972156078009421878487760, 5.04830879340312489077256240451, 5.95707716840111331779889935720, 6.77929753371211214142665085071, 7.23500500523022194032034749185, 7.898226305255516065182057931104

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.237146265\)
\(L(\frac12)\)	\(\approx\)	\(1.237146265\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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