LMFDB - L-function 2-450-5.4-c5-0-14

Dirichlet series

L(s) = 1

− 4i·2^-s − 16·4^-s + 98i·7^-s + 64i·8^-s − 354·11^-s − 404i·13^-s + 392·14^-s + 256·16^-s − 654i·17^-s − 1.79e3·19^-s + 1.41e3i·22^-s − 1.08e3i·23^-s − 1.61e3·26^-s − 1.56e3i·28^-s − 5.75e3·29^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s + 0.755i·7^-s + 0.353i·8^-s − 0.882·11^-s − 0.663i·13^-s + 0.534·14^-s + 0.250·16^-s − 0.548i·17^-s − 1.14·19^-s + 0.623i·22^-s − 0.425i·23^-s − 0.468·26^-s − 0.377i·28^-s − 1.27·29^-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

$L(3)$	$\approx$	$1.487134853$
$L(\frac12)$	$\approx$	$1.487134853$
$L(\frac{7}{2})$		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 - 98iT - 1.68e4T^{2} $
	11	$ 1 + 354T + 1.61e5T^{2} $
	13	$ 1 + 404iT - 3.71e5T^{2} $
	17	$ 1 + 654iT - 1.41e6T^{2} $
	19	$ 1 + 1.79e3T + 2.47e6T^{2} $
	23	$ 1 + 1.08e3iT - 6.43e6T^{2} $
	29	$ 1 + 5.75e3T + 2.05e7T^{2} $
	31	$ 1 - 1.01e4T + 2.86e7T^{2} $
	37	$ 1 - 5.55e3iT - 6.93e7T^{2} $
	41	$ 1 - 1.29e4T + 1.15e8T^{2} $
	43	$ 1 - 8.96e3iT - 1.47e8T^{2} $
	47	$ 1 - 5.40e3iT - 2.29e8T^{2} $
	53	$ 1 - 8.21e3iT - 4.18e8T^{2} $
	59	$ 1 - 3.95e3T + 7.14e8T^{2} $
	61	$ 1 - 962T + 8.44e8T^{2} $
	67	$ 1 + 1.79e4iT - 1.35e9T^{2} $
	71	$ 1 - 5.61e4T + 1.80e9T^{2} $
	73	$ 1 - 8.56e4iT - 2.07e9T^{2} $
	79	$ 1 - 2.60e4T + 3.07e9T^{2} $
	83	$ 1 + 9.34e4iT - 3.93e9T^{2} $
	89	$ 1 - 7.34e4T + 5.58e9T^{2} $
	97	$ 1 - 1.28e5iT - 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

−10.32201929136535336802421754363, −9.442535322036218021477214214738, −8.486584387266337560833474699133, −7.75318129817252636597864626224, −6.30745054784354111580924889557, −5.33840409146129457554554047888, −4.37541871248185439060241844236, −2.95222965365990008370619645064, −2.24496994235425324259953094475, −0.67309541194531818486015002203, 0.56070629357527069293556852313, 2.14348545924013479149918571766, 3.75244587907017742849056603563, 4.59940060664626995630977527685, 5.75348664454641273635208613074, 6.69354897610083707626986067754, 7.58171582189101263195110343002, 8.353678078345244855121736809398, 9.364117298900427358986604649774, 10.34219346670654745362660129872

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(1.487134853\)
\(L(\frac12)\)	\(\approx\)	\(1.487134853\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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