LMFDB - L-function 2-4056-13.12-c1-0-35

Dirichlet series

L(s) = 1

− 3^-s + 0.801i·5^-s − 1.69i·7^-s + 9^-s + 1.24i·11^-s − 0.801i·15^-s + 0.445·17^-s − 2.04i·19^-s + 1.69i·21^-s − 0.801·23^-s + 4.35·25^-s − 27^-s − 2.46·29^-s + 1.57i·31^-s − 1.24i·33^-s + ⋯

L(s) = 1

− 0.577·3^-s + 0.358i·5^-s − 0.639i·7^-s + 0.333·9^-s + 0.375i·11^-s − 0.207i·15^-s + 0.107·17^-s − 0.470i·19^-s + 0.369i·21^-s − 0.167·23^-s + 0.871·25^-s − 0.192·27^-s − 0.458·29^-s + 0.283i·31^-s − 0.217i·33^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$4056$ = $2^{3} \cdot 3 \cdot 13^{2}$
Sign:	$0.969 + 0.246i$
Analytic conductor:	$32.3873$
Root analytic conductor:	$5.69098$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4056} (337, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 4056,\ (\ :1/2),\ 0.969 + 0.246i)$

Particular Values

$L(1)$	$\approx$	$1.417460123$
$L(\frac12)$	$\approx$	$1.417460123$
$L(\frac{3}{2})$		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 + T $
	13	$ 1 $
good	5	$ 1 - 0.801iT - 5T^{2} $
	7	$ 1 + 1.69iT - 7T^{2} $
	11	$ 1 - 1.24iT - 11T^{2} $
	17	$ 1 - 0.445T + 17T^{2} $
	19	$ 1 + 2.04iT - 19T^{2} $
	23	$ 1 + 0.801T + 23T^{2} $
	29	$ 1 + 2.46T + 29T^{2} $
	31	$ 1 - 1.57iT - 31T^{2} $
	37	$ 1 - 9.54iT - 37T^{2} $
	41	$ 1 + 7.56iT - 41T^{2} $
	43	$ 1 - 0.286T + 43T^{2} $
	47	$ 1 - 0.542iT - 47T^{2} $
	53	$ 1 + 2.45T + 53T^{2} $
	59	$ 1 + 14.3iT - 59T^{2} $
	61	$ 1 - 7.92T + 61T^{2} $
	67	$ 1 - 2.81iT - 67T^{2} $
	71	$ 1 + 6.96iT - 71T^{2} $
	73	$ 1 - 3.69iT - 73T^{2} $
	79	$ 1 + 10.9T + 79T^{2} $
	83	$ 1 - 4.55iT - 83T^{2} $
	89	$ 1 - 8.02iT - 89T^{2} $
	97	$ 1 + 9.39iT - 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

−8.359275177454613372462282843689, −7.47588974023231798062159158688, −6.92121057847629432867217176718, −6.35185420721763894215591392965, −5.36103596510323088710918812840, −4.71110434843028911529113617483, −3.87216412542661647489383998254, −2.97621189790235926624988921606, −1.82439946168559576505279788239, −0.64981638818780446862334205694, 0.76904369826039086753600387934, 1.92294352551537288800895329023, 2.97141764790594780333519197526, 3.99346444618664180955658158972, 4.78837626923165773523859045246, 5.67926427429313110184670895890, 5.97740179258901074424721212153, 7.00440654379778224299911623728, 7.69666428323759063261177594447, 8.595525985663833800017496275945

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.417460123\)
\(L(\frac12)\)	\(\approx\)	\(1.417460123\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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