LMFDB - L-function 2-15e2-5.4-c9-0-48

Dirichlet series

L(s) = 1

− 8i·2^-s + 448·4^-s − 4.24e3i·7^-s − 7.68e3i·8^-s + 4.62e4·11^-s − 1.15e5i·13^-s − 3.39e4·14^-s + 1.67e5·16^-s + 4.94e5i·17^-s + 1.00e6·19^-s − 3.69e5i·22^-s + 5.32e5i·23^-s − 9.27e5·26^-s − 1.90e6i·28^-s + 4.19e6·29^-s + ⋯

L(s) = 1

− 0.353i·2^-s + 0.875·4^-s − 0.667i·7^-s − 0.662i·8^-s + 0.951·11^-s − 1.12i·13^-s − 0.236·14^-s + 0.640·16^-s + 1.43i·17^-s + 1.77·19^-s − 0.336i·22^-s + 0.396i·23^-s − 0.398·26^-s − 0.584i·28^-s + 1.10·29^-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

$L(5)$	$\approx$	$3.497219318$
$L(\frac12)$	$\approx$	$3.497219318$
$L(\frac{11}{2})$		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 + 8iT - 512T^{2} $
	7	$ 1 + 4.24e3iT - 4.03e7T^{2} $
	11	$ 1 - 4.62e4T + 2.35e9T^{2} $
	13	$ 1 + 1.15e5iT - 1.06e10T^{2} $
	17	$ 1 - 4.94e5iT - 1.18e11T^{2} $
	19	$ 1 - 1.00e6T + 3.22e11T^{2} $
	23	$ 1 - 5.32e5iT - 1.80e12T^{2} $
	29	$ 1 - 4.19e6T + 1.45e13T^{2} $
	31	$ 1 + 3.36e6T + 2.64e13T^{2} $
	37	$ 1 - 1.49e7iT - 1.29e14T^{2} $
	41	$ 1 + 1.10e7T + 3.27e14T^{2} $
	43	$ 1 + 6.39e6iT - 5.02e14T^{2} $
	47	$ 1 + 3.55e7iT - 1.11e15T^{2} $
	53	$ 1 + 3.97e7iT - 3.29e15T^{2} $
	59	$ 1 + 8.51e7T + 8.66e15T^{2} $
	61	$ 1 - 4.57e7T + 1.16e16T^{2} $
	67	$ 1 - 4.52e7iT - 2.72e16T^{2} $
	71	$ 1 - 1.89e8T + 4.58e16T^{2} $
	73	$ 1 - 4.12e8iT - 5.88e16T^{2} $
	79	$ 1 + 9.50e7T + 1.19e17T^{2} $
	83	$ 1 + 2.61e8iT - 1.86e17T^{2} $
	89	$ 1 + 1.99e7T + 3.50e17T^{2} $
	97	$ 1 - 1.95e7iT - 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

−10.41618685203474641564729696054, −9.856682873173891546414151676333, −8.378079643833706037168489072892, −7.39076986459748199919223785780, −6.51625539762276836186983257932, −5.42924106587393935270688719965, −3.82633071801641473964452032950, −3.09581896445654816887262996679, −1.58731637744054193507023015002, −0.839951539128751505509947826510, 1.04578085834865064059145459345, 2.20626429794735590276619415745, 3.24985485896328969263309143402, 4.79209843210303753298041364808, 5.88183936088126800927324284375, 6.83820752619558531912282011266, 7.56080385670561699302571687780, 8.939836001238696543381872643105, 9.587521466750334879493372082615, 11.03294601046388015777586446647

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 $\F_{q}$

\(L(5)\)	\(\approx\)	\(3.497219318\)
\(L(\frac12)\)	\(\approx\)	\(3.497219318\)
\(L(\frac{11}{2})\)		not available
\(L(1)\)		not available

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