LMFDB - L-function 2-3168-88.43-c1-0-21

Dirichlet series

L(s) = 1

− 2i·5^-s − 0.936·7^-s + (3.09 + 1.19i)11^-s + 4.27·13^-s + 3.33i·17^-s + 2.89i·19^-s + 5.12i·23^-s + 25^-s − 6.60·29^-s + 1.73i·31^-s + 1.87i·35^-s − 6.18i·37^-s + 1.46i·41^-s + 10.3i·43^-s + 6i·47^-s + ⋯

L(s) = 1

− 0.894i·5^-s − 0.353·7^-s + (0.932 + 0.361i)11^-s + 1.18·13^-s + 0.808i·17^-s + 0.664i·19^-s + 1.06i·23^-s + 0.200·25^-s − 1.22·29^-s + 0.311i·31^-s + 0.316i·35^-s − 1.01i·37^-s + 0.228i·41^-s + 1.57i·43^-s + 0.875i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3168$ = $2^{5} \cdot 3^{2} \cdot 11$
Sign:	$0.884 - 0.466i$
Analytic conductor:	$25.2966$
Root analytic conductor:	$5.02957$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3168} (2287, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3168,\ (\ :1/2),\ 0.884 - 0.466i)$

Particular Values

$L(1)$	$\approx$	$1.850867509$
$L(\frac12)$	$\approx$	$1.850867509$
$L(\frac{3}{2})$		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 $
	11	$ 1 + (-3.09 - 1.19i)T $
good	5	$ 1 + 2iT - 5T^{2} $
	7	$ 1 + 0.936T + 7T^{2} $
	13	$ 1 - 4.27T + 13T^{2} $
	17	$ 1 - 3.33iT - 17T^{2} $
	19	$ 1 - 2.89iT - 19T^{2} $
	23	$ 1 - 5.12iT - 23T^{2} $
	29	$ 1 + 6.60T + 29T^{2} $
	31	$ 1 - 1.73iT - 31T^{2} $
	37	$ 1 + 6.18iT - 37T^{2} $
	41	$ 1 - 1.46iT - 41T^{2} $
	43	$ 1 - 10.3iT - 43T^{2} $
	47	$ 1 - 6iT - 47T^{2} $
	53	$ 1 + 8.24iT - 53T^{2} $
	59	$ 1 + 9.65T + 59T^{2} $
	61	$ 1 - 9.06T + 61T^{2} $
	67	$ 1 - 10.2T + 67T^{2} $
	71	$ 1 - 4.24iT - 71T^{2} $
	73	$ 1 - 13.2iT - 73T^{2} $
	79	$ 1 - 3.86T + 79T^{2} $
	83	$ 1 + 2.39iT - 83T^{2} $
	89	$ 1 - 3.47T + 89T^{2} $
	97	$ 1 - 11.3T + 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

−8.720909249270766169884917166140, −8.148694924363536139914302647148, −7.27200621499959668030634165970, −6.31087408422923864273019376771, −5.82116734962437204090324339267, −4.86694165575632560531188715895, −3.91389975816825625013739133460, −3.44771172225608755101100585235, −1.82385012106730570000373933504, −1.12217816829237669145323765787, 0.66663176879635249029339811664, 2.05909616162718705863261345192, 3.14399966658414402496515791549, 3.68046069833180237839931684551, 4.70288815138285138962597528707, 5.76425043798023047718191896170, 6.55234659380919833417416966716, 6.86673708704444352283966580699, 7.82129313328336558699543828782, 8.814519771811710024311911280429

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.850867509\)
\(L(\frac12)\)	\(\approx\)	\(1.850867509\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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