LMFDB - L-function 2-2100-5.4-c3-0-8

Dirichlet series

L(s) = 1

− 3i·3^-s − 7i·7^-s − 9·9^-s + 4·11^-s − 54i·13^-s − 14i·17^-s − 92·19^-s − 21·21^-s + 152i·23^-s + 27i·27^-s + 106·29^-s − 144·31^-s − 12i·33^-s + 158i·37^-s − 162·39^-s + ⋯

L(s) = 1

− 0.577i·3^-s − 0.377i·7^-s − 0.333·9^-s + 0.109·11^-s − 1.15i·13^-s − 0.199i·17^-s − 1.11·19^-s − 0.218·21^-s + 1.37i·23^-s + 0.192i·27^-s + 0.678·29^-s − 0.834·31^-s − 0.0633i·33^-s + 0.702i·37^-s − 0.665·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2100$ = $2^{2} \cdot 3 \cdot 5^{2} \cdot 7$
Sign:	$0.447 - 0.894i$
Analytic conductor:	$123.904$
Root analytic conductor:	$11.1312$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2100} (1849, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2100,\ (\ :3/2),\ 0.447 - 0.894i)$

Particular Values

$L(2)$	$\approx$	$0.8980327255$
$L(\frac12)$	$\approx$	$0.8980327255$
$L(\frac{5}{2})$		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 + 3iT $
	5	$ 1 $
	7	$ 1 + 7iT $
good	11	$ 1 - 4T + 1.33e3T^{2} $
	13	$ 1 + 54iT - 2.19e3T^{2} $
	17	$ 1 + 14iT - 4.91e3T^{2} $
	19	$ 1 + 92T + 6.85e3T^{2} $
	23	$ 1 - 152iT - 1.21e4T^{2} $
	29	$ 1 - 106T + 2.43e4T^{2} $
	31	$ 1 + 144T + 2.97e4T^{2} $
	37	$ 1 - 158iT - 5.06e4T^{2} $
	41	$ 1 + 390T + 6.89e4T^{2} $
	43	$ 1 - 508iT - 7.95e4T^{2} $
	47	$ 1 + 528iT - 1.03e5T^{2} $
	53	$ 1 + 606iT - 1.48e5T^{2} $
	59	$ 1 - 364T + 2.05e5T^{2} $
	61	$ 1 - 678T + 2.26e5T^{2} $
	67	$ 1 - 844iT - 3.00e5T^{2} $
	71	$ 1 + 8T + 3.57e5T^{2} $
	73	$ 1 - 422iT - 3.89e5T^{2} $
	79	$ 1 + 384T + 4.93e5T^{2} $
	83	$ 1 - 548iT - 5.71e5T^{2} $
	89	$ 1 + 1.19e3T + 7.04e5T^{2} $
	97	$ 1 + 1.50e3iT - 9.12e5T^{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

−8.542222737489697285052910572286, −8.245525450570863198102638050201, −7.23360785145816925042800614774, −6.70870919036987273729668193862, −5.71584198271652716078803442801, −5.03452139139164025086740393084, −3.85464774016382354694199052601, −3.03862808860091806089384428942, −1.91986023388424447428465672186, −0.893914474519227313754950317785, 0.20740427915651909944462589503, 1.78937730751274217184649329827, 2.64518240234891050233234077620, 3.86524136341126907758285643072, 4.45315358010991380576863299478, 5.34907356905472767499163831716, 6.32606465557654087179360601218, 6.86202624460057260219195230945, 8.014987727968908900509961073315, 8.873710292611585553740468772235

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(0.8980327255\)
\(L(\frac12)\)	\(\approx\)	\(0.8980327255\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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