LMFDB - L-function 2-5328-12.11-c1-0-69

Dirichlet series

L(s) = 1

− 2.30i·5^-s − 4.07i·7^-s + 2.75·11^-s − 4.51·13^-s − 3.51i·17^-s − 5.45i·19^-s − 0.877·23^-s − 0.293·25^-s + 0.290i·29^-s + 1.57i·31^-s − 9.38·35^-s − 37^-s − 3.40i·41^-s − 2.37i·43^-s − 0.815·47^-s + ⋯

L(s) = 1

− 1.02i·5^-s − 1.54i·7^-s + 0.831·11^-s − 1.25·13^-s − 0.852i·17^-s − 1.25i·19^-s − 0.183·23^-s − 0.0586·25^-s + 0.0539i·29^-s + 0.283i·31^-s − 1.58·35^-s − 0.164·37^-s − 0.532i·41^-s − 0.362i·43^-s − 0.118·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$5328$ = $2^{4} \cdot 3^{2} \cdot 37$
Sign:	$-0.995 - 0.0917i$
Analytic conductor:	$42.5442$
Root analytic conductor:	$6.52259$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{5328} (2591, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 5328,\ (\ :1/2),\ -0.995 - 0.0917i)$

Particular Values

$L(1)$	$\approx$	$1.365883313$
$L(\frac12)$	$\approx$	$1.365883313$
$L(\frac{3}{2})$		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 $
	37	$ 1 + T $
good	5	$ 1 + 2.30iT - 5T^{2} $
	7	$ 1 + 4.07iT - 7T^{2} $
	11	$ 1 - 2.75T + 11T^{2} $
	13	$ 1 + 4.51T + 13T^{2} $
	17	$ 1 + 3.51iT - 17T^{2} $
	19	$ 1 + 5.45iT - 19T^{2} $
	23	$ 1 + 0.877T + 23T^{2} $
	29	$ 1 - 0.290iT - 29T^{2} $
	31	$ 1 - 1.57iT - 31T^{2} $
	41	$ 1 + 3.40iT - 41T^{2} $
	43	$ 1 + 2.37iT - 43T^{2} $
	47	$ 1 + 0.815T + 47T^{2} $
	53	$ 1 + 11.8iT - 53T^{2} $
	59	$ 1 - 6.27T + 59T^{2} $
	61	$ 1 - 3.66T + 61T^{2} $
	67	$ 1 - 4.98iT - 67T^{2} $
	71	$ 1 + 10.7T + 71T^{2} $
	73	$ 1 - 11.0T + 73T^{2} $
	79	$ 1 - 10.8iT - 79T^{2} $
	83	$ 1 + 14.3T + 83T^{2} $
	89	$ 1 + 7.85iT - 89T^{2} $
	97	$ 1 - 14.8T + 97T^{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

−7.69832601400426514911667433109, −7.03183792944039690842488882627, −6.73775104216211676540560038428, −5.38088733181242050084109041953, −4.79904546592224611864917753029, −4.27266937114857716777694350138, −3.42001361201647731320118151255, −2.27138138347538367601067115901, −1.07032923967732251268312535344, −0.39120667320500375506535517938, 1.65024764572429843395765772987, 2.43562838440389138419345760143, 3.13654092580088441943933884940, 4.03446285750194150146978229394, 4.99710638234541755061813490374, 5.93131299336537299790919971800, 6.22256652468772534956727288574, 7.10257191313629758742940997173, 7.82741567773260674748297799793, 8.556288603156579700510599675162

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 $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(1.365883313\)
\(L(\frac12)\)	\(\approx\)	\(1.365883313\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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