LMFDB - L-function 2-2475-5.4-c1-0-2

Dirichlet series

L(s) = 1

+ 2.41i·2^-s − 3.82·4^-s + 2i·7^-s − 4.41i·8^-s − 11^-s − 1.17i·13^-s − 4.82·14^-s + 2.99·16^-s + 6.82i·17^-s − 2.41i·22^-s + 2.82i·23^-s + 2.82·26^-s − 7.65i·28^-s − 3.65·29^-s − 1.58i·32^-s + ⋯

L(s) = 1

+ 1.70i·2^-s − 1.91·4^-s + 0.755i·7^-s − 1.56i·8^-s − 0.301·11^-s − 0.324i·13^-s − 1.29·14^-s + 0.749·16^-s + 1.65i·17^-s − 0.514i·22^-s + 0.589i·23^-s + 0.554·26^-s − 1.44i·28^-s − 0.679·29^-s − 0.280i·32^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2475$ = $3^{2} \cdot 5^{2} \cdot 11$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$19.7629$
Root analytic conductor:	$4.44555$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2475} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2475,\ (\ :1/2),\ 0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$0.5120984901$
$L(\frac12)$	$\approx$	$0.5120984901$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 $
	11	$ 1 + T $
good	2	$ 1 - 2.41iT - 2T^{2} $
	7	$ 1 - 2iT - 7T^{2} $
	13	$ 1 + 1.17iT - 13T^{2} $
	17	$ 1 - 6.82iT - 17T^{2} $
	19	$ 1 + 19T^{2} $
	23	$ 1 - 2.82iT - 23T^{2} $
	29	$ 1 + 3.65T + 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 - 7.65iT - 37T^{2} $
	41	$ 1 + 6T + 41T^{2} $
	43	$ 1 + 6iT - 43T^{2} $
	47	$ 1 - 2.82iT - 47T^{2} $
	53	$ 1 + 11.6iT - 53T^{2} $
	59	$ 1 - 1.65T + 59T^{2} $
	61	$ 1 + 9.31T + 61T^{2} $
	67	$ 1 + 12.4iT - 67T^{2} $
	71	$ 1 + 11.3T + 71T^{2} $
	73	$ 1 + 1.17iT - 73T^{2} $
	79	$ 1 + 4T + 79T^{2} $
	83	$ 1 - 6iT - 83T^{2} $
	89	$ 1 + 13.3T + 89T^{2} $
	97	$ 1 + 3.65iT - 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

−9.207902377363908052170114763309, −8.423976457611660027020278036326, −8.104965376871287019274846062808, −7.21831158537682424248816285274, −6.43463135488561258033832122942, −5.75927646012269045906764173688, −5.25464414472577267861072262022, −4.29022071614590204580742344669, −3.27936586476421750957615978291, −1.81932044797704165512034690848, 0.17912477784042206012875641648, 1.26411635372000649165420672362, 2.41680025702599308989805076111, 3.13165290902666212124197552909, 4.12980074950403156154230096914, 4.67929527846158112829302915121, 5.66324912172298670937505015561, 6.96815568421542097967573567466, 7.56001271835610505330760725984, 8.722021174831330762511179097826

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.5120984901\)
\(L(\frac12)\)	\(\approx\)	\(0.5120984901\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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