LMFDB - L-function 2-84e2-28.27-c1-0-97

Dirichlet series

L(s) = 1

− 3.21i·5^-s − 1.36i·11^-s − 2.93i·13^-s − 6.91i·17^-s − 7.35·19^-s − 3.62i·23^-s − 5.33·25^-s + 1.11·29^-s − 8.70·31^-s + 7.63·37^-s + 0.833i·41^-s − 4.82i·43^-s + 2.95·47^-s + 4.57·53^-s − 4.39·55^-s + ⋯

L(s) = 1

− 1.43i·5^-s − 0.412i·11^-s − 0.812i·13^-s − 1.67i·17^-s − 1.68·19^-s − 0.756i·23^-s − 1.06·25^-s + 0.206·29^-s − 1.56·31^-s + 1.25·37^-s + 0.130i·41^-s − 0.735i·43^-s + 0.430·47^-s + 0.628·53^-s − 0.592·55^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$7056$ = $2^{4} \cdot 3^{2} \cdot 7^{2}$
Sign:	$-0.810 - 0.585i$
Analytic conductor:	$56.3424$
Root analytic conductor:	$7.50616$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{7056} (1567, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 7056,\ (\ :1/2),\ -0.810 - 0.585i)$

Particular Values

$L(1)$	$\approx$	$0.8928180255$
$L(\frac12)$	$\approx$	$0.8928180255$
$L(\frac{3}{2})$		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 $
good	5	$ 1 + 3.21iT - 5T^{2} $
	11	$ 1 + 1.36iT - 11T^{2} $
	13	$ 1 + 2.93iT - 13T^{2} $
	17	$ 1 + 6.91iT - 17T^{2} $
	19	$ 1 + 7.35T + 19T^{2} $
	23	$ 1 + 3.62iT - 23T^{2} $
	29	$ 1 - 1.11T + 29T^{2} $
	31	$ 1 + 8.70T + 31T^{2} $
	37	$ 1 - 7.63T + 37T^{2} $
	41	$ 1 - 0.833iT - 41T^{2} $
	43	$ 1 + 4.82iT - 43T^{2} $
	47	$ 1 - 2.95T + 47T^{2} $
	53	$ 1 - 4.57T + 53T^{2} $
	59	$ 1 + 14.0T + 59T^{2} $
	61	$ 1 - 11.0iT - 61T^{2} $
	67	$ 1 - 12.1iT - 67T^{2} $
	71	$ 1 - 12.6iT - 71T^{2} $
	73	$ 1 + 6.49iT - 73T^{2} $
	79	$ 1 + 7.79iT - 79T^{2} $
	83	$ 1 - 8.87T + 83T^{2} $
	89	$ 1 + 12.6iT - 89T^{2} $
	97	$ 1 - 13.8iT - 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

−7.63772633244298023993963653636, −6.88154608240927246333663983703, −5.90254193277536214209752271203, −5.42189935069954333960149723255, −4.61371979818428335536242943033, −4.15322787145109997555542043536, −3.00080467488892753469436342848, −2.16885156746215684565884441397, −0.966840967612186044873720634701, −0.23691610720906243972611928420, 1.74764038244705617405733688838, 2.22892581501558591715071492479, 3.32222149303470947376213610663, 3.94203558595023277165611037852, 4.63659662370893208533643759286, 5.86205603792677419190911316898, 6.34227259778825040035283727981, 6.84068808346347437158817189090, 7.63503600795211668179085006268, 8.183392252499578751428327669271

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8928180255\)
\(L(\frac12)\)	\(\approx\)	\(0.8928180255\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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