LMFDB - L-function 2-15e2-5.4-c9-0-49

Dirichlet series

L(s) = 1

− 21.4i·2^-s + 53.2·4^-s + 9.90e3i·7^-s − 1.21e4i·8^-s − 3.64e4·11^-s − 1.64e5i·13^-s + 2.12e5·14^-s − 2.32e5·16^-s − 8.23e4i·17^-s + 6.09e5·19^-s + 7.80e5i·22^-s + 1.88e6i·23^-s − 3.53e6·26^-s + 5.27e5i·28^-s + 3.39e5·29^-s + ⋯

L(s) = 1

− 0.946i·2^-s + 0.103·4^-s + 1.55i·7^-s − 1.04i·8^-s − 0.750·11^-s − 1.60i·13^-s + 1.47·14^-s − 0.885·16^-s − 0.239i·17^-s + 1.07·19^-s + 0.710i·22^-s + 1.40i·23^-s − 1.51·26^-s + 0.162i·28^-s + 0.0890·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$225$ = $3^{2} \cdot 5^{2}$
Sign:	$-0.894 + 0.447i$
Analytic conductor:	$115.883$
Root analytic conductor:	$10.7648$
Motivic weight:	$9$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{225} (199, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 225,\ (\ :9/2),\ -0.894 + 0.447i)$

Particular Values

$L(5)$	$\approx$	$1.666609069$
$L(\frac12)$	$\approx$	$1.666609069$
$L(\frac{11}{2})$		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 $
good	2	$ 1 + 21.4iT - 512T^{2} $
	7	$ 1 - 9.90e3iT - 4.03e7T^{2} $
	11	$ 1 + 3.64e4T + 2.35e9T^{2} $
	13	$ 1 + 1.64e5iT - 1.06e10T^{2} $
	17	$ 1 + 8.23e4iT - 1.18e11T^{2} $
	19	$ 1 - 6.09e5T + 3.22e11T^{2} $
	23	$ 1 - 1.88e6iT - 1.80e12T^{2} $
	29	$ 1 - 3.39e5T + 1.45e13T^{2} $
	31	$ 1 - 5.47e5T + 2.64e13T^{2} $
	37	$ 1 - 5.25e6iT - 1.29e14T^{2} $
	41	$ 1 + 2.05e6T + 3.27e14T^{2} $
	43	$ 1 - 6.76e6iT - 5.02e14T^{2} $
	47	$ 1 + 3.15e7iT - 1.11e15T^{2} $
	53	$ 1 + 4.89e7iT - 3.29e15T^{2} $
	59	$ 1 - 8.77e7T + 8.66e15T^{2} $
	61	$ 1 - 3.84e7T + 1.16e16T^{2} $
	67	$ 1 + 1.36e8iT - 2.72e16T^{2} $
	71	$ 1 + 3.49e8T + 4.58e16T^{2} $
	73	$ 1 + 1.61e8iT - 5.88e16T^{2} $
	79	$ 1 - 1.26e8T + 1.19e17T^{2} $
	83	$ 1 + 2.87e8iT - 1.86e17T^{2} $
	89	$ 1 + 5.63e8T + 3.50e17T^{2} $
	97	$ 1 + 4.71e8iT - 7.60e17T^{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

−10.22942793780627070506691031290, −9.544657328747035668318518919851, −8.352876679913468890358466981017, −7.34104611043548222888789873388, −5.84522537444177853391238388801, −5.18116717214589291539852180404, −3.28949724990134381274324889217, −2.76663106215943924718595896436, −1.65279819035062547416512195426, −0.34958379675838115518790244956, 1.10031449984650016470891852110, 2.46376861918762420813662163323, 4.01479829323062086452700255149, 4.96247274055776040214507280723, 6.29550616419551925282100144707, 7.09664917126110521795974435336, 7.71556411966186527616886334829, 8.840160406288248596752235767562, 10.13424023624837246169014456970, 10.94151620735450039016147371984

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(1.666609069\)
\(L(\frac12)\)	\(\approx\)	\(1.666609069\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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