LMFDB - L-function 2-1008-28.27-c3-0-59

Dirichlet series

L(s) = 1

− 22.0i·5^-s + 18.5·7^-s − 58.3i·11^-s − 22.0i·17^-s − 153.·19^-s − 174. i·23^-s − 360.·25^-s + 201.·31^-s − 408. i·35^-s + 376·37^-s + 330. i·41^-s + 343.·49^-s − 1.28e3·55^-s + 1.10e3i·71^-s − 1.08e3i·77^-s + ⋯

L(s) = 1

− 1.97i·5^-s + 0.999·7^-s − 1.59i·11^-s − 0.314i·17^-s − 1.85·19^-s − 1.58i·23^-s − 2.88·25^-s + 1.16·31^-s − 1.97i·35^-s + 1.67·37^-s + 1.25i·41^-s + 1.00·49^-s − 3.15·55^-s + 1.85i·71^-s − 1.59i·77^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1008$ = $2^{4} \cdot 3^{2} \cdot 7$
Sign:	$-1$
Analytic conductor:	$59.4739$
Root analytic conductor:	$7.71193$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1008} (559, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1008,\ (\ :3/2),\ -1)$

Particular Values

$L(2)$	$\approx$	$1.717585857$
$L(\frac12)$	$\approx$	$1.717585857$
$L(\frac{5}{2})$		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 - 18.5T $
good	5	$ 1 + 22.0iT - 125T^{2} $
	11	$ 1 + 58.3iT - 1.33e3T^{2} $
	13	$ 1 - 2.19e3T^{2} $
	17	$ 1 + 22.0iT - 4.91e3T^{2} $
	19	$ 1 + 153.T + 6.85e3T^{2} $
	23	$ 1 + 174. iT - 1.21e4T^{2} $
	29	$ 1 + 2.43e4T^{2} $
	31	$ 1 - 201.T + 2.97e4T^{2} $
	37	$ 1 - 376T + 5.06e4T^{2} $
	41	$ 1 - 330. iT - 6.89e4T^{2} $
	43	$ 1 - 7.95e4T^{2} $
	47	$ 1 + 1.03e5T^{2} $
	53	$ 1 + 1.48e5T^{2} $
	59	$ 1 + 2.05e5T^{2} $
	61	$ 1 - 2.26e5T^{2} $
	67	$ 1 - 3.00e5T^{2} $
	71	$ 1 - 1.10e3iT - 3.57e5T^{2} $
	73	$ 1 - 3.89e5T^{2} $
	79	$ 1 - 4.93e5T^{2} $
	83	$ 1 + 5.71e5T^{2} $
	89	$ 1 - 639. iT - 7.04e5T^{2} $
	97	$ 1 - 9.12e5T^{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

−8.794097971247070088431634102929, −8.410880896284583765290614565467, −8.027781548837726028784071838098, −6.36570523296108509617143001592, −5.62918494932002685449546171469, −4.59597186222254579317475231781, −4.25418795928688909799377580137, −2.49322642021045713380158398998, −1.19137633460034693736002637358, −0.44362854031654256385628605427, 1.85847874238367326984168743169, 2.46815666279875331612243423589, 3.79536864730048110270081561832, 4.60769530615919350116760982939, 5.93272732174792600379788977252, 6.71643118330078022863637242988, 7.47076107792385956599094405164, 8.021659485614403608245858807562, 9.353838357391258724532582486149, 10.23317120723341948262712007939

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.717585857\)
\(L(\frac12)\)	\(\approx\)	\(1.717585857\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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