LMFDB - L-function 2-450-5.4-c5-0-16

Dirichlet series

L(s) = 1

− 4i·2^-s − 16·4^-s − 148i·7^-s + 64i·8^-s + 384·11^-s + 334i·13^-s − 592·14^-s + 256·16^-s + 576i·17^-s + 664·19^-s − 1.53e3i·22^-s + 3.84e3i·23^-s + 1.33e3·26^-s + 2.36e3i·28^-s − 96·29^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 1.14i·7^-s + 0.353i·8^-s + 0.956·11^-s + 0.548i·13^-s − 0.807·14^-s + 0.250·16^-s + 0.483i·17^-s + 0.421·19^-s − 0.676i·22^-s + 1.51i·23^-s + 0.387·26^-s + 0.570i·28^-s − 0.0211·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$450$ = $2 \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$72.1727$
Root analytic conductor:	$8.49545$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{450} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 450,\ (\ :5/2),\ 0.894 + 0.447i)$

Particular Values

$L(3)$	$\approx$	$1.954083505$
$L(\frac12)$	$\approx$	$1.954083505$
$L(\frac{7}{2})$		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 + 4iT $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 + 148iT - 1.68e4T^{2} $
	11	$ 1 - 384T + 1.61e5T^{2} $
	13	$ 1 - 334iT - 3.71e5T^{2} $
	17	$ 1 - 576iT - 1.41e6T^{2} $
	19	$ 1 - 664T + 2.47e6T^{2} $
	23	$ 1 - 3.84e3iT - 6.43e6T^{2} $
	29	$ 1 + 96T + 2.05e7T^{2} $
	31	$ 1 + 4.56e3T + 2.86e7T^{2} $
	37	$ 1 - 5.79e3iT - 6.93e7T^{2} $
	41	$ 1 + 6.72e3T + 1.15e8T^{2} $
	43	$ 1 - 1.48e4iT - 1.47e8T^{2} $
	47	$ 1 + 1.92e4iT - 2.29e8T^{2} $
	53	$ 1 + 7.77e3iT - 4.18e8T^{2} $
	59	$ 1 - 1.30e4T + 7.14e8T^{2} $
	61	$ 1 - 4.27e4T + 8.44e8T^{2} $
	67	$ 1 - 3.66e4iT - 1.35e9T^{2} $
	71	$ 1 - 6.45e4T + 1.80e9T^{2} $
	73	$ 1 - 1.68e4iT - 2.07e9T^{2} $
	79	$ 1 + 2.80e4T + 3.07e9T^{2} $
	83	$ 1 - 6.64e4iT - 3.93e9T^{2} $
	89	$ 1 - 8.17e4T + 5.58e9T^{2} $
	97	$ 1 + 2.99e4iT - 8.58e9T^{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

−10.16490397046606604756889362883, −9.574956874725974396088810053022, −8.573932417044861998537832914963, −7.45449009513509772158924320010, −6.60592916217333221512572463878, −5.27179300142151465265446504070, −4.05447922885184289198486501316, −3.48491408541956880535222258337, −1.80362342114274887238597520393, −0.908637679731059634143987254474, 0.60743799720323446139953730640, 2.22754804753113614030135066722, 3.53894523194962432745134551621, 4.82873368931448336159888392019, 5.72597110602211587191814040087, 6.56057659951772148883116520480, 7.56145422401294351932073296518, 8.676441893540621789367844496240, 9.124578571066427613644576643936, 10.17032535483729759996765050339

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(1.954083505\)
\(L(\frac12)\)	\(\approx\)	\(1.954083505\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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