LMFDB - L-function 2-4032-3.2-c2-0-32

Dirichlet series

L(s) = 1

− 6.06i·5^-s + 2.64·7^-s + 12.1i·11^-s + 18.5·13^-s + 10.9i·17^-s + 20·19^-s − 12.1i·23^-s − 11.8·25^-s + 41.8i·29^-s − 25.1·31^-s − 16.0i·35^-s − 38·37^-s + 60.6i·41^-s + 83.4·43^-s − 16.9i·47^-s + ⋯

L(s) = 1

− 1.21i·5^-s + 0.377·7^-s + 1.10i·11^-s + 1.42·13^-s + 0.641i·17^-s + 1.05·19^-s − 0.527i·23^-s − 0.473·25^-s + 1.44i·29^-s − 0.811·31^-s − 0.458i·35^-s − 1.02·37^-s + 1.47i·41^-s + 1.94·43^-s − 0.361i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$0.577 - 0.816i$
Analytic conductor:	$109.864$
Root analytic conductor:	$10.4816$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1),\ 0.577 - 0.816i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.139207984$
$L(\frac12)$	$\approx$	$2.139207984$
$L(2)$		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 $
	7	$ 1 - 2.64T $
good	5	$ 1 + 6.06iT - 25T^{2} $
	11	$ 1 - 12.1iT - 121T^{2} $
	13	$ 1 - 18.5T + 169T^{2} $
	17	$ 1 - 10.9iT - 289T^{2} $
	19	$ 1 - 20T + 361T^{2} $
	23	$ 1 + 12.1iT - 529T^{2} $
	29	$ 1 - 41.8iT - 841T^{2} $
	31	$ 1 + 25.1T + 961T^{2} $
	37	$ 1 + 38T + 1.36e3T^{2} $
	41	$ 1 - 60.6iT - 1.68e3T^{2} $
	43	$ 1 - 83.4T + 1.84e3T^{2} $
	47	$ 1 + 16.9iT - 2.20e3T^{2} $
	53	$ 1 - 94.0iT - 2.80e3T^{2} $
	59	$ 1 - 58.2iT - 3.48e3T^{2} $
	61	$ 1 + 15.6T + 3.72e3T^{2} $
	67	$ 1 + 132.T + 4.48e3T^{2} $
	71	$ 1 + 12.1iT - 5.04e3T^{2} $
	73	$ 1 + 76.9T + 5.32e3T^{2} $
	79	$ 1 + 33.6T + 6.24e3T^{2} $
	83	$ 1 - 60.5iT - 6.88e3T^{2} $
	89	$ 1 - 4.77iT - 7.92e3T^{2} $
	97	$ 1 + 188.T + 9.40e3T^{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

−8.565707779679489814741862795507, −7.68615032288188991790056898563, −7.05863929574689888004932869919, −5.99234843284374819843538625265, −5.38851427443199277742332786341, −4.57052517176904588317512767045, −4.02272248479765831122598394032, −2.92379970393931530463906995580, −1.50422618934674504633642138726, −1.20902143821111564929458022069, 0.46745352451799692988423603390, 1.63683184973450382741403182108, 2.81355552975966932568241250751, 3.42102364047562669315266185016, 4.13182198154737023485698961602, 5.51154450748733726848728650862, 5.83412657631317526934748421161, 6.75229006406319693972211878931, 7.39586755592881394596801772174, 8.085664147441702079838183965151

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.139207984\)
\(L(\frac12)\)	\(\approx\)	\(2.139207984\)
\(L(2)\)		not available
\(L(1)\)		not available

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