LMFDB - L-function 2-1800-24.11-c1-0-58

Dirichlet series

L(s) = 1

− 1.41·2^-s + 2.00·4^-s − 5.16i·7^-s − 2.82·8^-s − 3.05i·11^-s − 0.837i·13^-s + 7.30i·14^-s + 4.00·16^-s + 6.32·19^-s + 4.32i·22^-s + 4.47·23^-s + 1.18i·26^-s − 10.3i·28^-s − 5.65·32^-s − 11.1i·37^-s − 8.94·38^-s + ⋯

L(s) = 1

− 1.00·2^-s + 1.00·4^-s − 1.95i·7^-s − 1.00·8^-s − 0.921i·11^-s − 0.232i·13^-s + 1.95i·14^-s + 1.00·16^-s + 1.45·19^-s + 0.921i·22^-s + 0.932·23^-s + 0.232i·26^-s − 1.95i·28^-s − 1.00·32^-s − 1.83i·37^-s − 1.45·38^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1800$ = $2^{3} \cdot 3^{2} \cdot 5^{2}$
Sign:	$-0.577 + 0.816i$
Analytic conductor:	$14.3730$
Root analytic conductor:	$3.79118$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1800} (251, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1800,\ (\ :1/2),\ -0.577 + 0.816i)$

Particular Values

$L(1)$	$\approx$	$0.9655629481$
$L(\frac12)$	$\approx$	$0.9655629481$
$L(\frac{3}{2})$		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 + 1.41T $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 + 5.16iT - 7T^{2} $
	11	$ 1 + 3.05iT - 11T^{2} $
	13	$ 1 + 0.837iT - 13T^{2} $
	17	$ 1 - 17T^{2} $
	19	$ 1 - 6.32T + 19T^{2} $
	23	$ 1 - 4.47T + 23T^{2} $
	29	$ 1 + 29T^{2} $
	31	$ 1 - 31T^{2} $
	37	$ 1 + 11.1iT - 37T^{2} $
	41	$ 1 - 10.3iT - 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 + 2.82T + 47T^{2} $
	53	$ 1 + 5.65T + 53T^{2} $
	59	$ 1 + 5.42iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 + 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 + 73T^{2} $
	79	$ 1 - 79T^{2} $
	83	$ 1 - 83T^{2} $
	89	$ 1 + 18.8iT - 89T^{2} $
	97	$ 1 + 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

−9.103138203630208912716367110605, −8.056527159416222445413331876395, −7.54483192015978878034719119740, −6.91766381468426018051435440779, −6.05014280417916190231927497585, −4.94851805815765072186320386818, −3.67983068518136028476191708739, −3.03290172324580880231259833930, −1.34525409574998873273209889783, −0.54736513321356428308095469775, 1.48796161635212817254516608465, 2.46395301339936813126659631037, 3.22568557039741587274800301970, 4.94369009632821361577309620330, 5.59918641660101664521998367870, 6.51660598102429233750799022721, 7.27958471947982021016977905132, 8.168234641294034667108220009688, 8.872763478527191938157188603592, 9.451579110810755810315854495967

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.9655629481\)
\(L(\frac12)\)	\(\approx\)	\(0.9655629481\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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