LMFDB - L-function 2-4032-168.125-c1-0-48

Dirichlet series

L(s) = 1

− 1.21i·5^-s + (−0.355 + 2.62i)7^-s − 4.06·11^-s + 1.42·13^-s + 4.74·17^-s − 4.84·19^-s − 6.03i·23^-s + 3.51·25^-s − 3.45·29^-s − 3.15i·31^-s + (3.19 + 0.432i)35^-s + 9.24i·37^-s + 1.24·41^-s + 3.23i·43^-s − 9.43·47^-s + ⋯

L(s) = 1

− 0.544i·5^-s + (−0.134 + 0.990i)7^-s − 1.22·11^-s + 0.396·13^-s + 1.15·17^-s − 1.11·19^-s − 1.25i·23^-s + 0.703·25^-s − 0.641·29^-s − 0.566i·31^-s + (0.539 + 0.0731i)35^-s + 1.51i·37^-s + 0.194·41^-s + 0.492i·43^-s − 1.37·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$-0.469 + 0.883i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (1889, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ -0.469 + 0.883i)$

Particular Values

$L(1)$	$\approx$	$0.8047931434$
$L(\frac12)$	$\approx$	$0.8047931434$
$L(\frac{3}{2})$		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 + (0.355 - 2.62i)T $
good	5	$ 1 + 1.21iT - 5T^{2} $
	11	$ 1 + 4.06T + 11T^{2} $
	13	$ 1 - 1.42T + 13T^{2} $
	17	$ 1 - 4.74T + 17T^{2} $
	19	$ 1 + 4.84T + 19T^{2} $
	23	$ 1 + 6.03iT - 23T^{2} $
	29	$ 1 + 3.45T + 29T^{2} $
	31	$ 1 + 3.15iT - 31T^{2} $
	37	$ 1 - 9.24iT - 37T^{2} $
	41	$ 1 - 1.24T + 41T^{2} $
	43	$ 1 - 3.23iT - 43T^{2} $
	47	$ 1 + 9.43T + 47T^{2} $
	53	$ 1 - 2.44T + 53T^{2} $
	59	$ 1 + 10.2iT - 59T^{2} $
	61	$ 1 + 10.1T + 61T^{2} $
	67	$ 1 + 6.51iT - 67T^{2} $
	71	$ 1 - 2.14iT - 71T^{2} $
	73	$ 1 + 12.6iT - 73T^{2} $
	79	$ 1 - 10.6T + 79T^{2} $
	83	$ 1 + 16.3iT - 83T^{2} $
	89	$ 1 - 14.9T + 89T^{2} $
	97	$ 1 + 11.4iT - 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

−8.163428856057764225469448109833, −7.80006591779177219903467279508, −6.48462197057830347117884835948, −6.05594080035129470894038772908, −5.06245026371734678993608110232, −4.71909400600835632284172309819, −3.40119625050673735119740341585, −2.65831877663432313392325844435, −1.69163903573784003930260582168, −0.23928451312791489292721629604, 1.16365212826125576618515872607, 2.38721647512335798402863191576, 3.35646994635338888743755528808, 3.92700928743899363483325801850, 5.02416484947140722174524832458, 5.69686637185045279462877612020, 6.56764030269718031604827325259, 7.37316521051478518883674185849, 7.73285278310924631383684557300, 8.578986476565727962327570457201

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8047931434\)
\(L(\frac12)\)	\(\approx\)	\(0.8047931434\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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