LMFDB - L-function 2-684-19.18-c2-0-3

Dirichlet series

L(s) = 1

− 0.556·5^-s − 9.52·7^-s + 13.5·11^-s − 10.5i·13^-s − 2.63·17^-s + (0.112 + 18.9i)19^-s + 22.0·23^-s − 24.6·25^-s + 48.7i·29^-s − 0.248i·31^-s + 5.30·35^-s + 59.3i·37^-s + 69.0i·41^-s − 27.5·43^-s − 44.8·47^-s + ⋯

L(s) = 1

− 0.111·5^-s − 1.36·7^-s + 1.22·11^-s − 0.810i·13^-s − 0.155·17^-s + (0.00593 + 0.999i)19^-s + 0.957·23^-s − 0.987·25^-s + 1.68i·29^-s − 0.00800i·31^-s + 0.151·35^-s + 1.60i·37^-s + 1.68i·41^-s − 0.641·43^-s − 0.955·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00593 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.00593 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$684$ = $2^{2} \cdot 3^{2} \cdot 19$
Sign:	$-0.00593 - 0.999i$
Analytic conductor:	$18.6376$
Root analytic conductor:	$4.31713$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{684} (37, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 684,\ (\ :1),\ -0.00593 - 0.999i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.078339068$
$L(\frac12)$	$\approx$	$1.078339068$
$L(2)$		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 $
	19	$ 1 + (-0.112 - 18.9i)T $
good	5	$ 1 + 0.556T + 25T^{2} $
	7	$ 1 + 9.52T + 49T^{2} $
	11	$ 1 - 13.5T + 121T^{2} $
	13	$ 1 + 10.5iT - 169T^{2} $
	17	$ 1 + 2.63T + 289T^{2} $
	23	$ 1 - 22.0T + 529T^{2} $
	29	$ 1 - 48.7iT - 841T^{2} $
	31	$ 1 + 0.248iT - 961T^{2} $
	37	$ 1 - 59.3iT - 1.36e3T^{2} $
	41	$ 1 - 69.0iT - 1.68e3T^{2} $
	43	$ 1 + 27.5T + 1.84e3T^{2} $
	47	$ 1 + 44.8T + 2.20e3T^{2} $
	53	$ 1 + 3.85iT - 2.80e3T^{2} $
	59	$ 1 - 59.3iT - 3.48e3T^{2} $
	61	$ 1 - 96.4T + 3.72e3T^{2} $
	67	$ 1 + 10.7iT - 4.48e3T^{2} $
	71	$ 1 + 52.6iT - 5.04e3T^{2} $
	73	$ 1 - 13.7T + 5.32e3T^{2} $
	79	$ 1 + 51.3iT - 6.24e3T^{2} $
	83	$ 1 - 63.7T + 6.88e3T^{2} $
	89	$ 1 - 44.9iT - 7.92e3T^{2} $
	97	$ 1 - 72.6iT - 9.40e3T^{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

−10.28561724765904665007059631580, −9.699691901202285181054016173632, −8.880759695318898678331978749829, −7.910516654493919655260932384290, −6.73934303229894247629896908142, −6.27058880839494454241216709000, −5.07549891864518168564946142177, −3.71353144095101397795874849645, −3.07495001873927676560506588429, −1.28113517333767567490516427401, 0.41582439145629959452990154355, 2.18789482347587048823303611013, 3.52782410408187123899258295794, 4.28083863619705193120467164634, 5.70768531931355133389985282145, 6.66586276929171537525412712229, 7.10362491122877989773806641703, 8.524671090259315409753306264696, 9.419336804723459696389355684828, 9.711968267348489095952709724010

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.078339068\)
\(L(\frac12)\)	\(\approx\)	\(1.078339068\)
\(L(2)\)		not available
\(L(1)\)		not available

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