LMFDB - L-function 2-15e2-5.4-c9-0-1

Dirichlet series

L(s) = 1

− 22i·2^-s + 28·4^-s − 5.98e3i·7^-s − 1.18e4i·8^-s + 1.46e4·11^-s − 3.79e4i·13^-s − 1.31e5·14^-s − 2.47e5·16^-s + 4.41e5i·17^-s − 4.41e5·19^-s − 3.22e5i·22^-s + 2.26e6i·23^-s − 8.33e5·26^-s − 1.67e5i·28^-s − 1.04e6·29^-s + ⋯

L(s) = 1

− 0.972i·2^-s + 0.0546·4^-s − 0.942i·7^-s − 1.02i·8^-s + 0.301·11^-s − 0.368i·13^-s − 0.916·14^-s − 0.942·16^-s + 1.28i·17^-s − 0.777·19^-s − 0.293i·22^-s + 1.68i·23^-s − 0.357·26^-s − 0.0515i·28^-s − 0.275·29^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$225$ = $3^{2} \cdot 5^{2}$
Sign:	$0.447 - 0.894i$
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:	$(2,\ 225,\ (\ :9/2),\ 0.447 - 0.894i)$

Particular Values

$L(5)$	$\approx$	$0.2888975303$
$L(\frac12)$	$\approx$	$0.2888975303$
$L(\frac{11}{2})$		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 + 22iT - 512T^{2} $
	7	$ 1 + 5.98e3iT - 4.03e7T^{2} $
	11	$ 1 - 1.46e4T + 2.35e9T^{2} $
	13	$ 1 + 3.79e4iT - 1.06e10T^{2} $
	17	$ 1 - 4.41e5iT - 1.18e11T^{2} $
	19	$ 1 + 4.41e5T + 3.22e11T^{2} $
	23	$ 1 - 2.26e6iT - 1.80e12T^{2} $
	29	$ 1 + 1.04e6T + 1.45e13T^{2} $
	31	$ 1 + 7.91e6T + 2.64e13T^{2} $
	37	$ 1 + 2.09e7iT - 1.29e14T^{2} $
	41	$ 1 + 1.32e7T + 3.27e14T^{2} $
	43	$ 1 - 2.31e7iT - 5.02e14T^{2} $
	47	$ 1 - 1.38e7iT - 1.11e15T^{2} $
	53	$ 1 + 5.76e7iT - 3.29e15T^{2} $
	59	$ 1 + 3.20e7T + 8.66e15T^{2} $
	61	$ 1 - 1.10e8T + 1.16e16T^{2} $
	67	$ 1 + 1.18e8iT - 2.72e16T^{2} $
	71	$ 1 + 2.76e8T + 4.58e16T^{2} $
	73	$ 1 - 2.64e8iT - 5.88e16T^{2} $
	79	$ 1 + 4.48e8T + 1.19e17T^{2} $
	83	$ 1 - 8.51e8iT - 1.86e17T^{2} $
	89	$ 1 - 1.89e8T + 3.50e17T^{2} $
	97	$ 1 + 1.01e9iT - 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

−10.84596692339171629293910255517, −10.12654490185219108987798355544, −9.141207662342788900242285890701, −7.76440111863353802031633541563, −6.88358508767655009286001259978, −5.70616481662662241234010215974, −4.03789626237209066435159452834, −3.51550199693124841666021900692, −2.02124372230029467650659929642, −1.19979607484612354070181669072, 0.05552853578543732343259639250, 1.88614505716930291086176720221, 2.84308566105805512912458925486, 4.56710931278375865737117193944, 5.54818740677476479596931532408, 6.49612687971401268647243655249, 7.25573203201931993515356773790, 8.508742896170141491667174171893, 9.041896229390973807230725744524, 10.43987571249715252952776585275

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(0.2888975303\)
\(L(\frac12)\)	\(\approx\)	\(0.2888975303\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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