LMFDB - L-function 2-450-5.4-c3-0-5

Dirichlet series

L(s) = 1

− 2i·2^-s − 4·4^-s − 14i·7^-s + 8i·8^-s − 6·11^-s + 68i·13^-s − 28·14^-s + 16·16^-s + 78i·17^-s − 44·19^-s + 12i·22^-s − 120i·23^-s + 136·26^-s + 56i·28^-s + 126·29^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 0.755i·7^-s + 0.353i·8^-s − 0.164·11^-s + 1.45i·13^-s − 0.534·14^-s + 0.250·16^-s + 1.11i·17^-s − 0.531·19^-s + 0.116i·22^-s − 1.08i·23^-s + 1.02·26^-s + 0.377i·28^-s + 0.806·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$450$ = $2 \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$26.5508$
Root analytic conductor:	$5.15275$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{450} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 450,\ (\ :3/2),\ 0.894 - 0.447i)$

Particular Values

$L(2)$	$\approx$	$1.303122503$
$L(\frac12)$	$\approx$	$1.303122503$
$L(\frac{5}{2})$		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 + 2iT $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 + 14iT - 343T^{2} $
	11	$ 1 + 6T + 1.33e3T^{2} $
	13	$ 1 - 68iT - 2.19e3T^{2} $
	17	$ 1 - 78iT - 4.91e3T^{2} $
	19	$ 1 + 44T + 6.85e3T^{2} $
	23	$ 1 + 120iT - 1.21e4T^{2} $
	29	$ 1 - 126T + 2.43e4T^{2} $
	31	$ 1 + 244T + 2.97e4T^{2} $
	37	$ 1 - 304iT - 5.06e4T^{2} $
	41	$ 1 - 480T + 6.89e4T^{2} $
	43	$ 1 - 104iT - 7.95e4T^{2} $
	47	$ 1 - 600iT - 1.03e5T^{2} $
	53	$ 1 - 258iT - 1.48e5T^{2} $
	59	$ 1 - 534T + 2.05e5T^{2} $
	61	$ 1 - 362T + 2.26e5T^{2} $
	67	$ 1 - 268iT - 3.00e5T^{2} $
	71	$ 1 - 972T + 3.57e5T^{2} $
	73	$ 1 - 470iT - 3.89e5T^{2} $
	79	$ 1 + 1.24e3T + 4.93e5T^{2} $
	83	$ 1 + 396iT - 5.71e5T^{2} $
	89	$ 1 + 972T + 7.04e5T^{2} $
	97	$ 1 - 46iT - 9.12e5T^{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

−10.79055908212195378412944646315, −10.02169034745124019765302137638, −9.042088177048726839310535444920, −8.199210416913976268165464284598, −7.02497890527338251595744714774, −6.06993041983229543660622433247, −4.53450362904211865588275318406, −3.95883014500040833368426778375, −2.45923174085190700919372007706, −1.19952123479345551092621614185, 0.46752539479720959482519751673, 2.44919589353584210765304264684, 3.74001390302376578162910707036, 5.28035131384639514186325059989, 5.64816651530711963779472048450, 6.99354451089136047503674363806, 7.79970490284372622912019963914, 8.727113518480559458849071489633, 9.515525246392198693287779669934, 10.50660331137782613243107151089

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.303122503\)
\(L(\frac12)\)	\(\approx\)	\(1.303122503\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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