LMFDB - L-function 2-15e2-15.14-c8-0-40

Dirichlet series

L(s) = 1

− 19.1·2^-s + 111.·4^-s − 367. i·7^-s + 2.76e3·8^-s − 9.36e3i·11^-s − 2.89e4i·13^-s + 7.04e3i·14^-s − 8.16e4·16^-s − 5.17e4·17^-s + 6.69e4·19^-s + 1.79e5i·22^-s − 1.18e5·23^-s + 5.55e5i·26^-s − 4.10e4i·28^-s − 1.18e6i·29^-s + ⋯

L(s) = 1

− 1.19·2^-s + 0.436·4^-s − 0.153i·7^-s + 0.675·8^-s − 0.639i·11^-s − 1.01i·13^-s + 0.183i·14^-s − 1.24·16^-s − 0.619·17^-s + 0.513·19^-s + 0.766i·22^-s − 0.422·23^-s + 1.21i·26^-s − 0.0667i·28^-s − 1.67i·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(9-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$225$ = $3^{2} \cdot 5^{2}$
Sign:	$-0.988 - 0.151i$
Analytic conductor:	$91.6601$
Root analytic conductor:	$9.57393$
Motivic weight:	$8$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{225} (224, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 225,\ (\ :4),\ -0.988 - 0.151i)$

Particular Values

$L(\frac{9}{2})$	$\approx$	$0.3012999235$
$L(\frac12)$	$\approx$	$0.3012999235$
$L(5)$		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 + 19.1T + 256T^{2} $
	7	$ 1 + 367. iT - 5.76e6T^{2} $
	11	$ 1 + 9.36e3iT - 2.14e8T^{2} $
	13	$ 1 + 2.89e4iT - 8.15e8T^{2} $
	17	$ 1 + 5.17e4T + 6.97e9T^{2} $
	19	$ 1 - 6.69e4T + 1.69e10T^{2} $
	23	$ 1 + 1.18e5T + 7.83e10T^{2} $
	29	$ 1 + 1.18e6iT - 5.00e11T^{2} $
	31	$ 1 + 2.31e5T + 8.52e11T^{2} $
	37	$ 1 - 1.63e6iT - 3.51e12T^{2} $
	41	$ 1 + 2.74e6iT - 7.98e12T^{2} $
	43	$ 1 - 1.44e6iT - 1.16e13T^{2} $
	47	$ 1 + 6.90e6T + 2.38e13T^{2} $
	53	$ 1 - 1.26e7T + 6.22e13T^{2} $
	59	$ 1 + 4.74e6iT - 1.46e14T^{2} $
	61	$ 1 - 7.53e6T + 1.91e14T^{2} $
	67	$ 1 + 2.64e7iT - 4.06e14T^{2} $
	71	$ 1 + 3.33e7iT - 6.45e14T^{2} $
	73	$ 1 - 4.47e7iT - 8.06e14T^{2} $
	79	$ 1 - 7.76e7T + 1.51e15T^{2} $
	83	$ 1 - 2.37e7T + 2.25e15T^{2} $
	89	$ 1 - 5.22e5iT - 3.93e15T^{2} $
	97	$ 1 - 3.64e7iT - 7.83e15T^{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

−10.16429546124983362429083386566, −9.315460256191296044274874447770, −8.300816519592606137971248052606, −7.70147696974344705350643254337, −6.45913005848700916795107731528, −5.18171591112100154693838690863, −3.81703394816765463606494677495, −2.34437054602188666461951655406, −0.948974463197960138711333614910, −0.13125906922870180722556230227, 1.26932290621879839060891503680, 2.24978612691605791631173518370, 4.00623341942945601758884532765, 5.11946780544717943605535323645, 6.71232143601716954119019286408, 7.44548161279864890988268658584, 8.623057504255302708898379514389, 9.259295027436686360625494030854, 10.12079521025316612412736189531, 11.05407194245840297900309300694

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(\frac{9}{2})\)	\(\approx\)	\(0.3012999235\)
\(L(\frac12)\)	\(\approx\)	\(0.3012999235\)
\(L(5)\)		not available
\(L(1)\)		not available

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