LMFDB - L-function 2-9900-5.4-c1-0-48

Dirichlet series

L(s) = 1

− 4.60i·7^-s − 11^-s − 4.60i·13^-s + 6.60i·17^-s + 7.21·19^-s + 8·29^-s + 9.21·31^-s + 3.21i·37^-s − 8·41^-s − 3.39i·43^-s − 5.21i·47^-s − 14.2·49^-s − 2i·53^-s + 8·59^-s + 7.21·61^-s + ⋯

L(s) = 1

− 1.74i·7^-s − 0.301·11^-s − 1.27i·13^-s + 1.60i·17^-s + 1.65·19^-s + 1.48·29^-s + 1.65·31^-s + 0.527i·37^-s − 1.24·41^-s − 0.517i·43^-s − 0.760i·47^-s − 2.03·49^-s − 0.274i·53^-s + 1.04·59^-s + 0.923·61^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$9900$ = $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$79.0518$
Root analytic conductor:	$8.89111$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{9900} (5149, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 9900,\ (\ :1/2),\ 0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$2.298829701$
$L(\frac12)$	$\approx$	$2.298829701$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
	11	$ 1 + T $
good	7	$ 1 + 4.60iT - 7T^{2} $
	13	$ 1 + 4.60iT - 13T^{2} $
	17	$ 1 - 6.60iT - 17T^{2} $
	19	$ 1 - 7.21T + 19T^{2} $
	23	$ 1 - 23T^{2} $
	29	$ 1 - 8T + 29T^{2} $
	31	$ 1 - 9.21T + 31T^{2} $
	37	$ 1 - 3.21iT - 37T^{2} $
	41	$ 1 + 8T + 41T^{2} $
	43	$ 1 + 3.39iT - 43T^{2} $
	47	$ 1 + 5.21iT - 47T^{2} $
	53	$ 1 + 2iT - 53T^{2} $
	59	$ 1 - 8T + 59T^{2} $
	61	$ 1 - 7.21T + 61T^{2} $
	67	$ 1 - 4iT - 67T^{2} $
	71	$ 1 - 14.4T + 71T^{2} $
	73	$ 1 - 0.605iT - 73T^{2} $
	79	$ 1 - 11.2T + 79T^{2} $
	83	$ 1 - 10.6iT - 83T^{2} $
	89	$ 1 - 6T + 89T^{2} $
	97	$ 1 - 12.4iT - 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

−7.70064397786973904133767476045, −6.77818633031916213508815414049, −6.42246004516519163667955514519, −5.30653848973342163430984386279, −4.90197534016633377082783896888, −3.82182236048240985832930225014, −3.51059638381379685621736191110, −2.55471312536737335486824210736, −1.20284702301033936445697917392, −0.71916706054640485543823445284, 0.876920776062417562883858176260, 2.06397241797772370928374304152, 2.73324191888146563582573694480, 3.27835679722124433688943609983, 4.62699478437363553479218241671, 4.99399301375601032310390408506, 5.67080537026876497076135471599, 6.47594859776679445498153094098, 6.97769346038436074424310273540, 7.86463279247759604382674720224

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.298829701\)
\(L(\frac12)\)	\(\approx\)	\(2.298829701\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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