LMFDB - L-function 2-90e2-5.4-c1-0-49

Dirichlet series

L(s) = 1

− 2.73i·7^-s + 5.52·11^-s − 3.52i·13^-s + 5.52i·17^-s − 7.52·19^-s + 0.734i·23^-s + 4.46·29^-s + 7.52·31^-s − 6.05i·37^-s − 1.05·41^-s − 3.52i·43^-s − 1.20i·47^-s − 0.475·49^-s + 1.46i·53^-s + 1.46·59^-s + ⋯

L(s) = 1

− 1.03i·7^-s + 1.66·11^-s − 0.977i·13^-s + 1.33i·17^-s − 1.72·19^-s + 0.153i·23^-s + 0.829·29^-s + 1.35·31^-s − 0.995i·37^-s − 0.164·41^-s − 0.537i·43^-s − 0.176i·47^-s − 0.0679·49^-s + 0.201i·53^-s + 0.191·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8100$ = $2^{2} \cdot 3^{4} \cdot 5^{2}$
Sign:	$0.447 + 0.894i$
Analytic conductor:	$64.6788$
Root analytic conductor:	$8.04231$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{8100} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 8100,\ (\ :1/2),\ 0.447 + 0.894i)$

Particular Values

$L(1)$	$\approx$	$2.162477998$
$L(\frac12)$	$\approx$	$2.162477998$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
good	7	$ 1 + 2.73iT - 7T^{2} $
	11	$ 1 - 5.52T + 11T^{2} $
	13	$ 1 + 3.52iT - 13T^{2} $
	17	$ 1 - 5.52iT - 17T^{2} $
	19	$ 1 + 7.52T + 19T^{2} $
	23	$ 1 - 0.734iT - 23T^{2} $
	29	$ 1 - 4.46T + 29T^{2} $
	31	$ 1 - 7.52T + 31T^{2} $
	37	$ 1 + 6.05iT - 37T^{2} $
	41	$ 1 + 1.05T + 41T^{2} $
	43	$ 1 + 3.52iT - 43T^{2} $
	47	$ 1 + 1.20iT - 47T^{2} $
	53	$ 1 - 1.46iT - 53T^{2} $
	59	$ 1 - 1.46T + 59T^{2} $
	61	$ 1 - 9.05T + 61T^{2} $
	67	$ 1 - 4.25iT - 67T^{2} $
	71	$ 1 - 10.0T + 71T^{2} $
	73	$ 1 - 8iT - 73T^{2} $
	79	$ 1 + 2T + 79T^{2} $
	83	$ 1 + 5.26iT - 83T^{2} $
	89	$ 1 - 3T + 89T^{2} $
	97	$ 1 + 9.46iT - 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

−7.71846938587072898088666210859, −6.86646689863074812583858602575, −6.42429928554086625101462543932, −5.81342521912369508757466601001, −4.70388755121975295661790901512, −3.95720331573963157716933526896, −3.72345749310668525117654778830, −2.46725659572926371875930681708, −1.45740992899267213214022018005, −0.61470772624640970451984914452, 0.962445464219422230315436149234, 2.03458113690953743936420451630, 2.67502342939639790112821420728, 3.70939381246043385141419305477, 4.51589157652304843666698458567, 4.96872058824924822256739912496, 6.20662334964125835714839471762, 6.44526788585065236876352432819, 7.01066543584293462772871350064, 8.161690966690211410119475985558

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.162477998\)
\(L(\frac12)\)	\(\approx\)	\(2.162477998\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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