LMFDB - L-function 2-2200-5.4-c1-0-16

Dirichlet series

L(s) = 1

− 3i·3^-s + i·7^-s − 6·9^-s − 11^-s + 6i·13^-s + 3i·17^-s + 5·19^-s + 3·21^-s + 2i·23^-s + 9i·27^-s + 5·29^-s + 5·31^-s + 3i·33^-s − i·37^-s + 18·39^-s + ⋯

L(s) = 1

− 1.73i·3^-s + 0.377i·7^-s − 2·9^-s − 0.301·11^-s + 1.66i·13^-s + 0.727i·17^-s + 1.14·19^-s + 0.654·21^-s + 0.417i·23^-s + 1.73i·27^-s + 0.928·29^-s + 0.898·31^-s + 0.522i·33^-s − 0.164i·37^-s + 2.88·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2200$ = $2^{3} \cdot 5^{2} \cdot 11$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$17.5670$
Root analytic conductor:	$4.19131$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2200} (1849, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2200,\ (\ :1/2),\ 0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.577445671$
$L(\frac12)$	$\approx$	$1.577445671$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
	11	$ 1 + T $
good	3	$ 1 + 3iT - 3T^{2} $
	7	$ 1 - iT - 7T^{2} $
	13	$ 1 - 6iT - 13T^{2} $
	17	$ 1 - 3iT - 17T^{2} $
	19	$ 1 - 5T + 19T^{2} $
	23	$ 1 - 2iT - 23T^{2} $
	29	$ 1 - 5T + 29T^{2} $
	31	$ 1 - 5T + 31T^{2} $
	37	$ 1 + iT - 37T^{2} $
	41	$ 1 + 2T + 41T^{2} $
	43	$ 1 + 12iT - 43T^{2} $
	47	$ 1 + 2iT - 47T^{2} $
	53	$ 1 - 13iT - 53T^{2} $
	59	$ 1 + 2T + 59T^{2} $
	61	$ 1 - T + 61T^{2} $
	67	$ 1 - 16iT - 67T^{2} $
	71	$ 1 - 15T + 71T^{2} $
	73	$ 1 + 10iT - 73T^{2} $
	79	$ 1 + 2T + 79T^{2} $
	83	$ 1 - 14iT - 83T^{2} $
	89	$ 1 + 9T + 89T^{2} $
	97	$ 1 + 16iT - 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

−8.750033814417047677490073171817, −8.176928184099292688683819116561, −7.27593001474265680545030941718, −6.83140352184636110640920369648, −6.04408279225911624459946297569, −5.29988444815431184312514503835, −4.05277105742667363201017698168, −2.78311894664760681471181884053, −1.97797179684993739565466728093, −1.06243452360794404199500209003, 0.65938615874655851966813234629, 2.86372061890973911105229442305, 3.24423617244256912430813827723, 4.35952353234606578331305780195, 5.05373846444936009992893334883, 5.58233092891280996574353017873, 6.68399549609787930413143766249, 7.903598124703250062304337594555, 8.316409529617214672824183153004, 9.438511322031159363848841503090

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.577445671\)
\(L(\frac12)\)	\(\approx\)	\(1.577445671\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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