LMFDB - L-function 2-1008-84.83-c2-0-23

Dirichlet series

L(s) = 1

− 0.718·5^-s + (6.52 − 2.54i)7^-s + 13.1·11^-s − 12.0i·13^-s + 17.0·17^-s − 6.05·19^-s − 10.8·23^-s − 24.4·25^-s + 8.20i·29^-s − 36.6·31^-s + (−4.68 + 1.82i)35^-s + 45.0·37^-s + 62.4·41^-s − 12.0i·43^-s − 87.3i·47^-s + ⋯

L(s) = 1

− 0.143·5^-s + (0.931 − 0.362i)7^-s + 1.19·11^-s − 0.926i·13^-s + 1.00·17^-s − 0.318·19^-s − 0.473·23^-s − 0.979·25^-s + 0.282i·29^-s − 1.18·31^-s + (−0.133 + 0.0521i)35^-s + 1.21·37^-s + 1.52·41^-s − 0.279i·43^-s − 1.85i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1008$ = $2^{4} \cdot 3^{2} \cdot 7$
Sign:	$0.719 + 0.694i$
Analytic conductor:	$27.4660$
Root analytic conductor:	$5.24080$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1008} (1007, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1008,\ (\ :1),\ 0.719 + 0.694i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.197592342$
$L(\frac12)$	$\approx$	$2.197592342$
$L(2)$		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 $
	7	$ 1 + (-6.52 + 2.54i)T $
good	5	$ 1 + 0.718T + 25T^{2} $
	11	$ 1 - 13.1T + 121T^{2} $
	13	$ 1 + 12.0iT - 169T^{2} $
	17	$ 1 - 17.0T + 289T^{2} $
	19	$ 1 + 6.05T + 361T^{2} $
	23	$ 1 + 10.8T + 529T^{2} $
	29	$ 1 - 8.20iT - 841T^{2} $
	31	$ 1 + 36.6T + 961T^{2} $
	37	$ 1 - 45.0T + 1.36e3T^{2} $
	41	$ 1 - 62.4T + 1.68e3T^{2} $
	43	$ 1 + 12.0iT - 1.84e3T^{2} $
	47	$ 1 + 87.3iT - 2.20e3T^{2} $
	53	$ 1 - 74.7iT - 2.80e3T^{2} $
	59	$ 1 + 22.0iT - 3.48e3T^{2} $
	61	$ 1 + 57.2iT - 3.72e3T^{2} $
	67	$ 1 - 47.6iT - 4.48e3T^{2} $
	71	$ 1 - 66.4T + 5.04e3T^{2} $
	73	$ 1 + 36.0iT - 5.32e3T^{2} $
	79	$ 1 + 46.6iT - 6.24e3T^{2} $
	83	$ 1 + 38.2iT - 6.88e3T^{2} $
	89	$ 1 - 126.T + 7.92e3T^{2} $
	97	$ 1 + 126. iT - 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

−9.688106865231397658292844911020, −8.805125877474677057744489857997, −7.87328579461630959024581010110, −7.39514194827369542927054698517, −6.15984095584759256830577344152, −5.38353278728088783947444253653, −4.24937559359368549039966748368, −3.50827429108442835735488453130, −1.97742183592923015207533209255, −0.807014666762139502994286146422, 1.22956921188608336611155312509, 2.25659790518499125778778713945, 3.80157756853500782343281221614, 4.45240965071664509589645257335, 5.64712345009108483234523940055, 6.38737750712499763229754361839, 7.52241635009207165751753931796, 8.120938544923347765608613624908, 9.197445651745133499074131020970, 9.601608146039476283609241579097

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 $\F_{q}$
Belyi maps

\(L(\frac{3}{2})\)	\(\approx\)	\(2.197592342\)
\(L(\frac12)\)	\(\approx\)	\(2.197592342\)
\(L(2)\)		not available
\(L(1)\)		not available

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