LMFDB - L-function 2-2200-5.4-c1-0-6

Dirichlet series

L(s) = 1

− 3i·3^-s + 2i·7^-s − 6·9^-s − 11^-s + 6i·17^-s − 4·19^-s + 6·21^-s + i·23^-s + 9i·27^-s + 8·29^-s − 7·31^-s + 3i·33^-s + i·37^-s + 4·41^-s + 6i·43^-s + ⋯

L(s) = 1

− 1.73i·3^-s + 0.755i·7^-s − 2·9^-s − 0.301·11^-s + 1.45i·17^-s − 0.917·19^-s + 1.30·21^-s + 0.208i·23^-s + 1.73i·27^-s + 1.48·29^-s − 1.25·31^-s + 0.522i·33^-s + 0.164i·37^-s + 0.624·41^-s + 0.914i·43^-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

$L(1)$	$\approx$	$1.046981847$
$L(\frac12)$	$\approx$	$1.046981847$
$L(\frac{3}{2})$		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 - 2iT - 7T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 - 6iT - 17T^{2} $
	19	$ 1 + 4T + 19T^{2} $
	23	$ 1 - iT - 23T^{2} $
	29	$ 1 - 8T + 29T^{2} $
	31	$ 1 + 7T + 31T^{2} $
	37	$ 1 - iT - 37T^{2} $
	41	$ 1 - 4T + 41T^{2} $
	43	$ 1 - 6iT - 43T^{2} $
	47	$ 1 - 8iT - 47T^{2} $
	53	$ 1 - 2iT - 53T^{2} $
	59	$ 1 - T + 59T^{2} $
	61	$ 1 - 4T + 61T^{2} $
	67	$ 1 - 5iT - 67T^{2} $
	71	$ 1 - 3T + 71T^{2} $
	73	$ 1 - 16iT - 73T^{2} $
	79	$ 1 + 2T + 79T^{2} $
	83	$ 1 + 2iT - 83T^{2} $
	89	$ 1 + 15T + 89T^{2} $
	97	$ 1 - 7iT - 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

−8.607736628451271902192117582668, −8.405694847337583240332319627412, −7.56383299680926231065162165924, −6.75829813671257692859733827703, −6.08008925938768878080423438816, −5.55537089147830317549722857115, −4.23647158915717059994512065236, −2.89315977376610542255111750131, −2.14580447153418266999785570821, −1.22377971679948036065245449994, 0.37836928282661036551724975602, 2.43578108182827528965647095380, 3.42080114523957512133246048914, 4.17553723749593452804927274421, 4.85929283512734445722717393183, 5.51155788214468242046076751569, 6.63468824755675029160922967536, 7.49184241046276286777154528319, 8.521906515164382943475503440682, 9.100300216802742681981262946203

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.046981847\)
\(L(\frac12)\)	\(\approx\)	\(1.046981847\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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