LMFDB - L-function 2-1008-84.83-c2-0-20

Dirichlet series

L(s) = 1

− 9.41·5^-s + (3.32 − 6.16i)7^-s + 17.6·11^-s + 17.9i·13^-s − 13.3·17^-s + 14.6·19^-s − 30.1·23^-s + 63.6·25^-s − 25.9i·29^-s + 11.5·31^-s + (−31.2 + 57.9i)35^-s − 17.9·37^-s − 12.6·41^-s − 19.2i·43^-s − 48.3i·47^-s + ⋯

L(s) = 1

− 1.88·5^-s + (0.474 − 0.880i)7^-s + 1.60·11^-s + 1.37i·13^-s − 0.787·17^-s + 0.770·19^-s − 1.31·23^-s + 2.54·25^-s − 0.894i·29^-s + 0.372·31^-s + (−0.893 + 1.65i)35^-s − 0.484·37^-s − 0.308·41^-s − 0.447i·43^-s − 1.02i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1008$ = $2^{4} \cdot 3^{2} \cdot 7$
Sign:	$-0.391 + 0.920i$
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:	$(2,\ 1008,\ (\ :1),\ -0.391 + 0.920i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.8399312516$
$L(\frac12)$	$\approx$	$0.8399312516$
$L(2)$		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 + (-3.32 + 6.16i)T $
good	5	$ 1 + 9.41T + 25T^{2} $
	11	$ 1 - 17.6T + 121T^{2} $
	13	$ 1 - 17.9iT - 169T^{2} $
	17	$ 1 + 13.3T + 289T^{2} $
	19	$ 1 - 14.6T + 361T^{2} $
	23	$ 1 + 30.1T + 529T^{2} $
	29	$ 1 + 25.9iT - 841T^{2} $
	31	$ 1 - 11.5T + 961T^{2} $
	37	$ 1 + 17.9T + 1.36e3T^{2} $
	41	$ 1 + 12.6T + 1.68e3T^{2} $
	43	$ 1 + 19.2iT - 1.84e3T^{2} $
	47	$ 1 + 48.3iT - 2.20e3T^{2} $
	53	$ 1 + 3.42iT - 2.80e3T^{2} $
	59	$ 1 - 53.9iT - 3.48e3T^{2} $
	61	$ 1 + 59.8iT - 3.72e3T^{2} $
	67	$ 1 + 104. iT - 4.48e3T^{2} $
	71	$ 1 + 76.5T + 5.04e3T^{2} $
	73	$ 1 + 142. iT - 5.32e3T^{2} $
	79	$ 1 - 47.9iT - 6.24e3T^{2} $
	83	$ 1 + 90.9iT - 6.88e3T^{2} $
	89	$ 1 + 122.T + 7.92e3T^{2} $
	97	$ 1 - 94.7iT - 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

−9.356618083034959613723691021993, −8.591980964810204268228334967148, −7.78756459044962692045193091587, −7.05147763361623374466900591307, −6.44761466201713743690127835666, −4.65671644916847506755940538557, −4.11741395253251189360043522399, −3.58027110725517179329638569059, −1.66655661393873353488825910204, −0.31227263236172267001743871145, 1.16951407725413659125542476124, 2.90767818580656691297296199027, 3.82155144732067973494283796909, 4.60612698791558353064156146049, 5.70207598123213143070439884851, 6.80743365339176355859821616230, 7.65009996107874339281827554324, 8.400822711112173385800187020962, 8.879778286269007148982391591016, 10.04485385400376829711282465751

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

\(L(\frac{3}{2})\)	\(\approx\)	\(0.8399312516\)
\(L(\frac12)\)	\(\approx\)	\(0.8399312516\)
\(L(2)\)		not available
\(L(1)\)		not available

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