LMFDB - L-function 2-4032-28.27-c1-0-75

Dirichlet series

L(s) = 1

− 3.33i·5^-s + (0.662 − 2.56i)7^-s − 5.03i·11^-s + 6.04i·13^-s − 5.20i·17^-s − 4.71·19^-s + 2.20i·23^-s − 6.12·25^-s − 7.24·29^-s + 6.04·31^-s + (−8.54 − 2.20i)35^-s − 5.12·37^-s − 5.20i·41^-s − 9.12i·43^-s + 3.74·47^-s + ⋯

L(s) = 1

− 1.49i·5^-s + (0.250 − 0.968i)7^-s − 1.51i·11^-s + 1.67i·13^-s − 1.26i·17^-s − 1.08·19^-s + 0.460i·23^-s − 1.22·25^-s − 1.34·29^-s + 1.08·31^-s + (−1.44 − 0.373i)35^-s − 0.842·37^-s − 0.813i·41^-s − 1.39i·43^-s + 0.546·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$-0.968 - 0.250i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (3583, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ -0.968 - 0.250i)$

Particular Values

$L(1)$	$\approx$	$1.108735255$
$L(\frac12)$	$\approx$	$1.108735255$
$L(\frac{3}{2})$		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 $
	7	$ 1 + (-0.662 + 2.56i)T $
good	5	$ 1 + 3.33iT - 5T^{2} $
	11	$ 1 + 5.03iT - 11T^{2} $
	13	$ 1 - 6.04iT - 13T^{2} $
	17	$ 1 + 5.20iT - 17T^{2} $
	19	$ 1 + 4.71T + 19T^{2} $
	23	$ 1 - 2.20iT - 23T^{2} $
	29	$ 1 + 7.24T + 29T^{2} $
	31	$ 1 - 6.04T + 31T^{2} $
	37	$ 1 + 5.12T + 37T^{2} $
	41	$ 1 + 5.20iT - 41T^{2} $
	43	$ 1 + 9.12iT - 43T^{2} $
	47	$ 1 - 3.74T + 47T^{2} $
	53	$ 1 + 1.58T + 53T^{2} $
	59	$ 1 - 13.3T + 59T^{2} $
	61	$ 1 + 12.8iT - 61T^{2} $
	67	$ 1 - 9.12iT - 67T^{2} $
	71	$ 1 - 12.2iT - 71T^{2} $
	73	$ 1 - 12.0iT - 73T^{2} $
	79	$ 1 - 5.12iT - 79T^{2} $
	83	$ 1 + 3.74T + 83T^{2} $
	89	$ 1 - 1.46iT - 89T^{2} $
	97	$ 1 + 6.78iT - 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

−8.339795694021554457294122101358, −7.25264653332882202438347821603, −6.71178093582347572238467050967, −5.62445502988073373791086183856, −5.04948753274711488938110025295, −4.14990515885343800733875107134, −3.74801430351113717445997928113, −2.23317589942724537897936039099, −1.19907962770698967732121141376, −0.32224147957398246491371238868, 1.81144008947689853166060341579, 2.51714521438618675542234076388, 3.25444535919765860278102866561, 4.25901228820294122783812164466, 5.17811188536910853137376486811, 6.11064560023017021506894920942, 6.47299245426329375421020234027, 7.48625054816608609234043878908, 7.954638192688351040943891285312, 8.730907842933032512135331411292

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.108735255\)
\(L(\frac12)\)	\(\approx\)	\(1.108735255\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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