LMFDB - L-function 2-528-44.43-c5-0-20

Dirichlet series

L(s) = 1

− 9i·3^-s + 4.87·5^-s − 228.·7^-s − 81·9^-s + (−241. − 320. i)11^-s + 402. i·13^-s − 43.8i·15^-s + 1.22e3i·17^-s − 984.·19^-s + 2.05e3i·21^-s + 1.68e3i·23^-s − 3.10e3·25^-s + 729i·27^-s + 534. i·29^-s − 6.04e3i·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 0.0871·5^-s − 1.76·7^-s − 0.333·9^-s + (−0.602 − 0.798i)11^-s + 0.661i·13^-s − 0.0503i·15^-s + 1.02i·17^-s − 0.625·19^-s + 1.01i·21^-s + 0.664i·23^-s − 0.992·25^-s + 0.192i·27^-s + 0.117i·29^-s − 1.12i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$528$ = $2^{4} \cdot 3 \cdot 11$
Sign:	$0.920 + 0.390i$
Analytic conductor:	$84.6826$
Root analytic conductor:	$9.20231$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{528} (175, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 528,\ (\ :5/2),\ 0.920 + 0.390i)$

Particular Values

$L(3)$	$\approx$	$0.9116964031$
$L(\frac12)$	$\approx$	$0.9116964031$
$L(\frac{7}{2})$		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 $
	3	$ 1 + 9iT $
	11	$ 1 + (241. + 320. i)T $
good	5	$ 1 - 4.87T + 3.12e3T^{2} $
	7	$ 1 + 228.T + 1.68e4T^{2} $
	13	$ 1 - 402. iT - 3.71e5T^{2} $
	17	$ 1 - 1.22e3iT - 1.41e6T^{2} $
	19	$ 1 + 984.T + 2.47e6T^{2} $
	23	$ 1 - 1.68e3iT - 6.43e6T^{2} $
	29	$ 1 - 534. iT - 2.05e7T^{2} $
	31	$ 1 + 6.04e3iT - 2.86e7T^{2} $
	37	$ 1 - 7.45e3T + 6.93e7T^{2} $
	41	$ 1 + 3.60e3iT - 1.15e8T^{2} $
	43	$ 1 + 2.90e3T + 1.47e8T^{2} $
	47	$ 1 + 322. iT - 2.29e8T^{2} $
	53	$ 1 - 5.19e3T + 4.18e8T^{2} $
	59	$ 1 - 4.30e4iT - 7.14e8T^{2} $
	61	$ 1 + 2.37e4iT - 8.44e8T^{2} $
	67	$ 1 + 1.69e4iT - 1.35e9T^{2} $
	71	$ 1 + 5.43e4iT - 1.80e9T^{2} $
	73	$ 1 - 5.84e3iT - 2.07e9T^{2} $
	79	$ 1 + 6.67e4T + 3.07e9T^{2} $
	83	$ 1 - 7.27e4T + 3.93e9T^{2} $
	89	$ 1 + 7.88e4T + 5.58e9T^{2} $
	97	$ 1 - 1.52e4T + 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

−9.931403004881078876835921392023, −9.170417455858525890102640257703, −8.198676283151256625099953967416, −7.21068950704919679713696962666, −6.17787635453913746278245492881, −5.84802476691720332907990581127, −4.06323886572784586266964646275, −3.13146838040895864864708668155, −2.02010057181231949498802642052, −0.46279789021418881785261719047, 0.43567326204067004741822178759, 2.45790069862087643813855111633, 3.25240743215210254463141042587, 4.39370212198634548214577457153, 5.48240323393657435160641812384, 6.43706791747378163209055287041, 7.30525945386633858979379237774, 8.479384866433694512891004106932, 9.565378404186081166467029901049, 9.946228234230910444244489390090

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(0.9116964031\)
\(L(\frac12)\)	\(\approx\)	\(0.9116964031\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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