LMFDB - L-function 2-4032-8.5-c1-0-24

Dirichlet series

L(s) = 1

+ 2.73i·5^-s + 7^-s + 6.19i·11^-s + 4.73·17^-s + 0.535i·19^-s + 5.26·23^-s − 2.46·25^-s − 3.46i·29^-s + 8.92·31^-s + 2.73i·35^-s + 10i·37^-s − 4.73·41^-s − 12.9i·43^-s + 6.92·47^-s + 49^-s + ⋯

L(s) = 1

+ 1.22i·5^-s + 0.377·7^-s + 1.86i·11^-s + 1.14·17^-s + 0.122i·19^-s + 1.09·23^-s − 0.492·25^-s − 0.643i·29^-s + 1.60·31^-s + 0.461i·35^-s + 1.64i·37^-s − 0.739·41^-s − 1.97i·43^-s + 1.01·47^-s + 0.142·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4032$ = $2^{6} \cdot 3^{2} \cdot 7$
Sign:	$-0.258 - 0.965i$
Analytic conductor:	$32.1956$
Root analytic conductor:	$5.67412$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4032} (2017, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4032,\ (\ :1/2),\ -0.258 - 0.965i)$

Particular Values

$L(1)$	$\approx$	$2.120556649$
$L(\frac12)$	$\approx$	$2.120556649$
$L(\frac{3}{2})$		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 - T $
good	5	$ 1 - 2.73iT - 5T^{2} $
	11	$ 1 - 6.19iT - 11T^{2} $
	13	$ 1 - 13T^{2} $
	17	$ 1 - 4.73T + 17T^{2} $
	19	$ 1 - 0.535iT - 19T^{2} $
	23	$ 1 - 5.26T + 23T^{2} $
	29	$ 1 + 3.46iT - 29T^{2} $
	31	$ 1 - 8.92T + 31T^{2} $
	37	$ 1 - 10iT - 37T^{2} $
	41	$ 1 + 4.73T + 41T^{2} $
	43	$ 1 + 12.9iT - 43T^{2} $
	47	$ 1 - 6.92T + 47T^{2} $
	53	$ 1 - 3.46iT - 53T^{2} $
	59	$ 1 - 2.92iT - 59T^{2} $
	61	$ 1 - 5.46iT - 61T^{2} $
	67	$ 1 + 0.535iT - 67T^{2} $
	71	$ 1 - 3.80T + 71T^{2} $
	73	$ 1 + 10.3T + 73T^{2} $
	79	$ 1 - 2.53T + 79T^{2} $
	83	$ 1 - 12.3iT - 83T^{2} $
	89	$ 1 - 4.73T + 89T^{2} $
	97	$ 1 + 14.3T + 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

−8.544238164624479043183743753189, −7.76903900194751734669364283733, −7.06574568360109192412322550533, −6.76480758391907683435642350639, −5.69837162815793733743422769822, −4.86226879844422622990943654673, −4.12431909850271278332690868730, −3.06297750811738445298997105722, −2.39894017068598488140019256689, −1.30734350205996572313445701465, 0.71466605925668100200645110581, 1.28920741656391630514287370364, 2.80813150945835635471898823782, 3.55624608294188957560913060744, 4.58567366077884012270527182137, 5.27241543775451807756315950856, 5.81376226581405424329276779281, 6.68688072455456636039734815255, 7.81586541594809919423000842980, 8.236585619146675614517323225458

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.120556649\)
\(L(\frac12)\)	\(\approx\)	\(2.120556649\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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