LMFDB - L-function 2-3528-3.2-c2-0-23

Dirichlet series

L(s) = 1

− 8.57i·5^-s + 14.5i·11^-s + 10.9·13^-s − 26.8i·17^-s − 7.97·19^-s + 26.8i·23^-s − 48.5·25^-s + 35.7i·29^-s + 23.8·31^-s − 38.3·37^-s + 27.8i·41^-s + 68.4·43^-s + 71.1i·47^-s + 63.4i·53^-s + 124.·55^-s + ⋯

L(s) = 1

− 1.71i·5^-s + 1.32i·11^-s + 0.838·13^-s − 1.57i·17^-s − 0.419·19^-s + 1.16i·23^-s − 1.94·25^-s + 1.23i·29^-s + 0.769·31^-s − 1.03·37^-s + 0.679i·41^-s + 1.59·43^-s + 1.51i·47^-s + 1.19i·53^-s + 2.26·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3528$ = $2^{3} \cdot 3^{2} \cdot 7^{2}$
Sign:	$0.816 - 0.577i$
Analytic conductor:	$96.1310$
Root analytic conductor:	$9.80464$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3528} (1961, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3528,\ (\ :1),\ 0.816 - 0.577i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.647153009$
$L(\frac12)$	$\approx$	$1.647153009$
$L(2)$		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 $
good	5	$ 1 + 8.57iT - 25T^{2} $
	11	$ 1 - 14.5iT - 121T^{2} $
	13	$ 1 - 10.9T + 169T^{2} $
	17	$ 1 + 26.8iT - 289T^{2} $
	19	$ 1 + 7.97T + 361T^{2} $
	23	$ 1 - 26.8iT - 529T^{2} $
	29	$ 1 - 35.7iT - 841T^{2} $
	31	$ 1 - 23.8T + 961T^{2} $
	37	$ 1 + 38.3T + 1.36e3T^{2} $
	41	$ 1 - 27.8iT - 1.68e3T^{2} $
	43	$ 1 - 68.4T + 1.84e3T^{2} $
	47	$ 1 - 71.1iT - 2.20e3T^{2} $
	53	$ 1 - 63.4iT - 2.80e3T^{2} $
	59	$ 1 + 8.92iT - 3.48e3T^{2} $
	61	$ 1 + 5.77T + 3.72e3T^{2} $
	67	$ 1 + 34.6T + 4.48e3T^{2} $
	71	$ 1 - 68.6iT - 5.04e3T^{2} $
	73	$ 1 + 131.T + 5.32e3T^{2} $
	79	$ 1 - 30.0T + 6.24e3T^{2} $
	83	$ 1 + 86.5iT - 6.88e3T^{2} $
	89	$ 1 - 128. iT - 7.92e3T^{2} $
	97	$ 1 - 80.0T + 9.40e3T^{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

−8.641431177719275865836404790189, −7.69885188128144124311148381702, −7.19933626755712422915333679461, −6.09567269437483485359785940415, −5.26484366604008587657120942032, −4.69740053996323436555981102340, −4.10091120245798068590240267722, −2.87127407300149821034916143218, −1.62668966083714909916055894058, −0.986034254357598902645277285895, 0.40018054732134283028943195084, 1.91509965730205200839309604302, 2.80321430035635409575897993449, 3.60129379999628763462268138133, 4.14658172112930527568180878887, 5.67161140660581432609482351009, 6.23643020228546288524533194550, 6.59868298699154225564931262293, 7.58496824860661638911084411628, 8.394914074433522080365085863449

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

\(L(\frac{3}{2})\)	\(\approx\)	\(1.647153009\)
\(L(\frac12)\)	\(\approx\)	\(1.647153009\)
\(L(2)\)		not available
\(L(1)\)		not available

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