LMFDB - L-function 2-30e2-5.4-c3-0-9

Dirichlet series

L(s) = 1

− 7i·7^-s + 54·11^-s + 55i·13^-s − 18i·17^-s + 25·19^-s − 18i·23^-s − 54·29^-s − 271·31^-s + 314i·37^-s + 360·41^-s + 163i·43^-s − 522i·47^-s + 294·49^-s − 36i·53^-s + 126·59^-s + ⋯

L(s) = 1

− 0.377i·7^-s + 1.48·11^-s + 1.17i·13^-s − 0.256i·17^-s + 0.301·19^-s − 0.163i·23^-s − 0.345·29^-s − 1.57·31^-s + 1.39i·37^-s + 1.37·41^-s + 0.578i·43^-s − 1.62i·47^-s + 0.857·49^-s − 0.0933i·53^-s + 0.278·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$900$ = $2^{2} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$53.1017$
Root analytic conductor:	$7.28709$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{900} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 900,\ (\ :3/2),\ 0.894 - 0.447i)$

Particular Values

$L(2)$	$\approx$	$2.196241152$
$L(\frac12)$	$\approx$	$2.196241152$
$L(\frac{5}{2})$		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 $
	5	$ 1 $
good	7	$ 1 + 7iT - 343T^{2} $
	11	$ 1 - 54T + 1.33e3T^{2} $
	13	$ 1 - 55iT - 2.19e3T^{2} $
	17	$ 1 + 18iT - 4.91e3T^{2} $
	19	$ 1 - 25T + 6.85e3T^{2} $
	23	$ 1 + 18iT - 1.21e4T^{2} $
	29	$ 1 + 54T + 2.43e4T^{2} $
	31	$ 1 + 271T + 2.97e4T^{2} $
	37	$ 1 - 314iT - 5.06e4T^{2} $
	41	$ 1 - 360T + 6.89e4T^{2} $
	43	$ 1 - 163iT - 7.95e4T^{2} $
	47	$ 1 + 522iT - 1.03e5T^{2} $
	53	$ 1 + 36iT - 1.48e5T^{2} $
	59	$ 1 - 126T + 2.05e5T^{2} $
	61	$ 1 - 47T + 2.26e5T^{2} $
	67	$ 1 + 343iT - 3.00e5T^{2} $
	71	$ 1 - 1.08e3T + 3.57e5T^{2} $
	73	$ 1 - 1.05e3iT - 3.89e5T^{2} $
	79	$ 1 - 568T + 4.93e5T^{2} $
	83	$ 1 - 1.42e3iT - 5.71e5T^{2} $
	89	$ 1 - 1.44e3T + 7.04e5T^{2} $
	97	$ 1 + 439iT - 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

−9.492121335054437903208358807526, −9.214571536136264920504379289044, −8.130800541329968481664752096672, −7.01019777164308631751311503509, −6.58937134875236234111564721511, −5.40316328573450888373305853268, −4.25460544231362368151426102862, −3.61310313425532991933151187177, −2.08055695867881172962423382344, −0.978311506715119554673637558918, 0.71826212018856001737328380193, 1.99150276417406948795029389390, 3.31631711913275901123032187752, 4.15261123177848659814572961117, 5.45849187789021591257034664522, 6.06772443958902467981983828011, 7.17845278542478162134890350893, 7.923206977798691821157488472972, 9.082140258860466943296266022024, 9.378385635052092194339047562209

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(2.196241152\)
\(L(\frac12)\)	\(\approx\)	\(2.196241152\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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