LMFDB - L-function 2-1008-21.20-c1-0-14

Dirichlet series

L(s) = 1

− 1.53·5^-s + (0.414 − 2.61i)7^-s + 0.585i·11^-s + 2.16i·13^-s − 5.86·17^-s − 5.22i·19^-s − 2.24i·23^-s − 2.65·25^-s + 5.41i·29^-s − 4.32i·31^-s + (−0.634 + 4i)35^-s + 4·37^-s − 8.92·41^-s − 10.4·43^-s − 7.39·47^-s + ⋯

L(s) = 1

− 0.684·5^-s + (0.156 − 0.987i)7^-s + 0.176i·11^-s + 0.600i·13^-s − 1.42·17^-s − 1.19i·19^-s − 0.467i·23^-s − 0.531·25^-s + 1.00i·29^-s − 0.777i·31^-s + (−0.107 + 0.676i)35^-s + 0.657·37^-s − 1.39·41^-s − 1.59·43^-s − 1.07·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1008$ = $2^{4} \cdot 3^{2} \cdot 7$
Sign:	$-0.896 + 0.442i$
Analytic conductor:	$8.04892$
Root analytic conductor:	$2.83706$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1008} (881, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1008,\ (\ :1/2),\ -0.896 + 0.442i)$

Particular Values

$L(1)$	$\approx$	$0.4810617634$
$L(\frac12)$	$\approx$	$0.4810617634$
$L(\frac{3}{2})$		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 + (-0.414 + 2.61i)T $
good	5	$ 1 + 1.53T + 5T^{2} $
	11	$ 1 - 0.585iT - 11T^{2} $
	13	$ 1 - 2.16iT - 13T^{2} $
	17	$ 1 + 5.86T + 17T^{2} $
	19	$ 1 + 5.22iT - 19T^{2} $
	23	$ 1 + 2.24iT - 23T^{2} $
	29	$ 1 - 5.41iT - 29T^{2} $
	31	$ 1 + 4.32iT - 31T^{2} $
	37	$ 1 - 4T + 37T^{2} $
	41	$ 1 + 8.92T + 41T^{2} $
	43	$ 1 + 10.4T + 43T^{2} $
	47	$ 1 + 7.39T + 47T^{2} $
	53	$ 1 + 5.41iT - 53T^{2} $
	59	$ 1 + 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 9.65T + 67T^{2} $
	71	$ 1 + 4.58iT - 71T^{2} $
	73	$ 1 + 12.6iT - 73T^{2} $
	79	$ 1 - 2.34T + 79T^{2} $
	83	$ 1 + 13.5T + 83T^{2} $
	89	$ 1 + 5.86T + 89T^{2} $
	97	$ 1 + 8.28iT - 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

−9.617624096074408835714448071269, −8.713401095882491135532029942028, −7.946873465704492659578605930478, −6.94858084343562091580231792265, −6.58188660656238793184164059022, −4.92271026265499184264855244526, −4.35768998077195623278183191858, −3.37297314119773595030376885592, −1.92216567264704060052244492666, −0.20935810779684221209552993690, 1.83994628920741909558490762796, 3.07314379990765318279080540854, 4.09582377129762790925337993451, 5.16218299409769242545182470597, 6.01827787505478317548527868390, 6.93837171342014713745268461700, 8.213171684945024129473622952594, 8.313470031083564203581368263680, 9.496519304934470175352751868317, 10.25949087841005958000339057767

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.4810617634\)
\(L(\frac12)\)	\(\approx\)	\(0.4810617634\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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