LMFDB - L-function 2-15e2-15.14-c8-0-28

Dirichlet series

L(s) = 1

− 7.31·2^-s − 202.·4^-s − 3.66e3i·7^-s + 3.35e3·8^-s + 1.32e4i·11^-s − 2.16e4i·13^-s + 2.67e4i·14^-s + 2.73e4·16^-s − 1.17e5·17^-s + 2.55e5·19^-s − 9.69e4i·22^-s + 4.21e5·23^-s + 1.58e5i·26^-s + 7.41e5i·28^-s + 6.10e5i·29^-s + ⋯

L(s) = 1

− 0.457·2^-s − 0.791·4^-s − 1.52i·7^-s + 0.818·8^-s + 0.905i·11^-s − 0.756i·13^-s + 0.697i·14^-s + 0.416·16^-s − 1.40·17^-s + 1.96·19^-s − 0.414i·22^-s + 1.50·23^-s + 0.345i·26^-s + 1.20i·28^-s + 0.863i·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(9-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$225$ = $3^{2} \cdot 5^{2}$
Sign:	$0.472 + 0.881i$
Analytic conductor:	$91.6601$
Root analytic conductor:	$9.57393$
Motivic weight:	$8$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{225} (224, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 225,\ (\ :4),\ 0.472 + 0.881i)$

Particular Values

$L(\frac{9}{2})$	$\approx$	$1.221398322$
$L(\frac12)$	$\approx$	$1.221398322$
$L(5)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 $
good	2	$ 1 + 7.31T + 256T^{2} $
	7	$ 1 + 3.66e3iT - 5.76e6T^{2} $
	11	$ 1 - 1.32e4iT - 2.14e8T^{2} $
	13	$ 1 + 2.16e4iT - 8.15e8T^{2} $
	17	$ 1 + 1.17e5T + 6.97e9T^{2} $
	19	$ 1 - 2.55e5T + 1.69e10T^{2} $
	23	$ 1 - 4.21e5T + 7.83e10T^{2} $
	29	$ 1 - 6.10e5iT - 5.00e11T^{2} $
	31	$ 1 + 9.80e5T + 8.52e11T^{2} $
	37	$ 1 - 3.00e5iT - 3.51e12T^{2} $
	41	$ 1 - 1.06e6iT - 7.98e12T^{2} $
	43	$ 1 - 1.60e6iT - 1.16e13T^{2} $
	47	$ 1 + 7.57e5T + 2.38e13T^{2} $
	53	$ 1 - 3.30e6T + 6.22e13T^{2} $
	59	$ 1 - 2.09e7iT - 1.46e14T^{2} $
	61	$ 1 - 2.16e7T + 1.91e14T^{2} $
	67	$ 1 + 1.46e7iT - 4.06e14T^{2} $
	71	$ 1 - 1.44e7iT - 6.45e14T^{2} $
	73	$ 1 + 2.16e7iT - 8.06e14T^{2} $
	79	$ 1 + 4.05e7T + 1.51e15T^{2} $
	83	$ 1 - 4.84e7T + 2.25e15T^{2} $
	89	$ 1 + 3.57e7iT - 3.93e15T^{2} $
	97	$ 1 + 6.01e7iT - 7.83e15T^{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

−10.45020992474919988601514119306, −9.688090984868640758088168010780, −8.796948373346305633639218674344, −7.48190310215305143870431235952, −7.10288763155451585470426381863, −5.21187831827776324176305301553, −4.40639751162529488341456397598, −3.27748980612682401617476467855, −1.36838273119111281656126842306, −0.52147074782766168404397718337, 0.75498254496132276991965592483, 2.17007717574057315143470593739, 3.48737102857937938930863483993, 4.93113312957978797581994610514, 5.69886045666985382345583841382, 7.06995861586216302253829612112, 8.390909120391465083206642572010, 9.045452040875430329575397366060, 9.560825386395882969771829343675, 11.06776737532426423126071807416

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 $\F_{q}$
Belyi maps

\(L(\frac{9}{2})\)	\(\approx\)	\(1.221398322\)
\(L(\frac12)\)	\(\approx\)	\(1.221398322\)
\(L(5)\)		not available
\(L(1)\)		not available

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