LMFDB - L-function 2-3042-13.12-c1-0-31

Dirichlet series

L(s) = 1

− i·2^-s − 4^-s − 0.692i·5^-s + 0.356i·7^-s + i·8^-s − 0.692·10^-s − 2.93i·11^-s + 0.356·14^-s + 16^-s + 6.71·17^-s + 7.20i·19^-s + 0.692i·20^-s − 2.93·22^-s + 2.39·23^-s + 4.52·25^-s + ⋯

L(s) = 1

− 0.707i·2^-s − 0.5·4^-s − 0.309i·5^-s + 0.134i·7^-s + 0.353i·8^-s − 0.218·10^-s − 0.886i·11^-s + 0.0953·14^-s + 0.250·16^-s + 1.62·17^-s + 1.65i·19^-s + 0.154i·20^-s − 0.626·22^-s + 0.499·23^-s + 0.904·25^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3042$ = $2 \cdot 3^{2} \cdot 13^{2}$
Sign:	$0.722 + 0.691i$
Analytic conductor:	$24.2904$
Root analytic conductor:	$4.92853$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3042} (1351, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3042,\ (\ :1/2),\ 0.722 + 0.691i)$

Particular Values

$L(1)$	$\approx$	$1.775327928$
$L(\frac12)$	$\approx$	$1.775327928$
$L(\frac{3}{2})$		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 + iT $
	3	$ 1 $
	13	$ 1 $
good	5	$ 1 + 0.692iT - 5T^{2} $
	7	$ 1 - 0.356iT - 7T^{2} $
	11	$ 1 + 2.93iT - 11T^{2} $
	17	$ 1 - 6.71T + 17T^{2} $
	19	$ 1 - 7.20iT - 19T^{2} $
	23	$ 1 - 2.39T + 23T^{2} $
	29	$ 1 + 7.82T + 29T^{2} $
	31	$ 1 - 2.76iT - 31T^{2} $
	37	$ 1 - 10.0iT - 37T^{2} $
	41	$ 1 - 4.89iT - 41T^{2} $
	43	$ 1 + 6.59T + 43T^{2} $
	47	$ 1 + 4.98iT - 47T^{2} $
	53	$ 1 - 8.88T + 53T^{2} $
	59	$ 1 + 1.64iT - 59T^{2} $
	61	$ 1 + 6.49T + 61T^{2} $
	67	$ 1 + 13.5iT - 67T^{2} $
	71	$ 1 - 6.81iT - 71T^{2} $
	73	$ 1 - 3.18iT - 73T^{2} $
	79	$ 1 - 15.0T + 79T^{2} $
	83	$ 1 + 14.8iT - 83T^{2} $
	89	$ 1 + 0.396iT - 89T^{2} $
	97	$ 1 - 0.417iT - 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

−8.584800679323670959578750349888, −8.139632560957381896095391331758, −7.28091444917212705143174312061, −6.11012111389510805458123640507, −5.50288706425919099100770591783, −4.75223344181142212354282833371, −3.48674390085005408833760264444, −3.24887704946416486598518241515, −1.79127807445756998702957369811, −0.912012950108303274995866563765, 0.77193529652993301878705525888, 2.24169481577823610567760601846, 3.31945734836349994017739508654, 4.21877636840781733290609689176, 5.13712785653246651092822889516, 5.68812764624503948902423858543, 6.77239934108980274182313572353, 7.30543914056683057481994797616, 7.74641508154614251581705269676, 8.939576099005923243167256619240

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.775327928\)
\(L(\frac12)\)	\(\approx\)	\(1.775327928\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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