LMFDB - L-function 2-1008-7.6-c2-0-33

Dirichlet series

L(s) = 1

− 2.74i·5^-s + (5.19 − 4.69i)7^-s − 7.83·11^-s + 2.62i·13^-s − 1.90i·17^-s − 26.1i·19^-s − 26.3·23^-s + 17.4·25^-s − 5.12·29^-s − 30.2i·31^-s + (−12.8 − 14.2i)35^-s + 28.3·37^-s + 42.5i·41^-s − 68.7·43^-s + 44.5i·47^-s + ⋯

L(s) = 1

− 0.549i·5^-s + (0.742 − 0.670i)7^-s − 0.712·11^-s + 0.202i·13^-s − 0.111i·17^-s − 1.37i·19^-s − 1.14·23^-s + 0.697·25^-s − 0.176·29^-s − 0.976i·31^-s + (−0.368 − 0.407i)35^-s + 0.765·37^-s + 1.03i·41^-s − 1.59·43^-s + 0.948i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.670i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1008$ = $2^{4} \cdot 3^{2} \cdot 7$
Sign:	$-0.742 + 0.670i$
Analytic conductor:	$27.4660$
Root analytic conductor:	$5.24080$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1008} (433, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1008,\ (\ :1),\ -0.742 + 0.670i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.214456945$
$L(\frac12)$	$\approx$	$1.214456945$
$L(2)$		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 + (-5.19 + 4.69i)T $
good	5	$ 1 + 2.74iT - 25T^{2} $
	11	$ 1 + 7.83T + 121T^{2} $
	13	$ 1 - 2.62iT - 169T^{2} $
	17	$ 1 + 1.90iT - 289T^{2} $
	19	$ 1 + 26.1iT - 361T^{2} $
	23	$ 1 + 26.3T + 529T^{2} $
	29	$ 1 + 5.12T + 841T^{2} $
	31	$ 1 + 30.2iT - 961T^{2} $
	37	$ 1 - 28.3T + 1.36e3T^{2} $
	41	$ 1 - 42.5iT - 1.68e3T^{2} $
	43	$ 1 + 68.7T + 1.84e3T^{2} $
	47	$ 1 - 44.5iT - 2.20e3T^{2} $
	53	$ 1 + 57.2T + 2.80e3T^{2} $
	59	$ 1 + 94.6iT - 3.48e3T^{2} $
	61	$ 1 - 33.9iT - 3.72e3T^{2} $
	67	$ 1 + 35.9T + 4.48e3T^{2} $
	71	$ 1 + 58.3T + 5.04e3T^{2} $
	73	$ 1 - 74.0iT - 5.32e3T^{2} $
	79	$ 1 + 22.0T + 6.24e3T^{2} $
	83	$ 1 + 129. iT - 6.88e3T^{2} $
	89	$ 1 + 135. iT - 7.92e3T^{2} $
	97	$ 1 + 29.6iT - 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

−9.495748377242401571486782046688, −8.495452096193010349999703686311, −7.87302567058023887053192094336, −7.04516838355716371702437729507, −5.97018922192870807332288071240, −4.83385918492418821944167388970, −4.41662671809171735000772791543, −2.97893113270645537332241979474, −1.69122493429215040536758641683, −0.36780902214151847618727004110, 1.63500613988551714693772643633, 2.68271710743899549878644858736, 3.78316612869764459392176843958, 5.01446656467422772532560204260, 5.73909303770588727011868456869, 6.67017924756326140255014818040, 7.80891004171348838193702004570, 8.236575909806542735159471785184, 9.230126421992073029249333523716, 10.35977682639491970612111324279

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.214456945\)
\(L(\frac12)\)	\(\approx\)	\(1.214456945\)
\(L(2)\)		not available
\(L(1)\)		not available

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