LMFDB - L-function 2-30e2-15.14-c2-0-6

Dirichlet series

L(s) = 1

− i·7^-s − 4.24i·11^-s + 7i·13^-s − 4.24·17^-s + 7·19^-s + 29.6·23^-s − 29.6i·29^-s + 17·31^-s − 16i·37^-s − 50.9i·41^-s + 55i·43^-s − 46.6·47^-s + 48·49^-s + 84.8·53^-s − 55.1i·59^-s + ⋯

L(s) = 1

− 0.142i·7^-s − 0.385i·11^-s + 0.538i·13^-s − 0.249·17^-s + 0.368·19^-s + 1.29·23^-s − 1.02i·29^-s + 0.548·31^-s − 0.432i·37^-s − 1.24i·41^-s + 1.27i·43^-s − 0.992·47^-s + 0.979·49^-s + 1.60·53^-s − 0.934i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$900$ = $2^{2} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.881 + 0.472i$
Analytic conductor:	$24.5232$
Root analytic conductor:	$4.95209$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{900} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 900,\ (\ :1),\ 0.881 + 0.472i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.840822557$
$L(\frac12)$	$\approx$	$1.840822557$
$L(2)$		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 + iT - 49T^{2} $
	11	$ 1 + 4.24iT - 121T^{2} $
	13	$ 1 - 7iT - 169T^{2} $
	17	$ 1 + 4.24T + 289T^{2} $
	19	$ 1 - 7T + 361T^{2} $
	23	$ 1 - 29.6T + 529T^{2} $
	29	$ 1 + 29.6iT - 841T^{2} $
	31	$ 1 - 17T + 961T^{2} $
	37	$ 1 + 16iT - 1.36e3T^{2} $
	41	$ 1 + 50.9iT - 1.68e3T^{2} $
	43	$ 1 - 55iT - 1.84e3T^{2} $
	47	$ 1 + 46.6T + 2.20e3T^{2} $
	53	$ 1 - 84.8T + 2.80e3T^{2} $
	59	$ 1 + 55.1iT - 3.48e3T^{2} $
	61	$ 1 - 65T + 3.72e3T^{2} $
	67	$ 1 + 49iT - 4.48e3T^{2} $
	71	$ 1 + 50.9iT - 5.04e3T^{2} $
	73	$ 1 - 88iT - 5.32e3T^{2} $
	79	$ 1 - 40T + 6.24e3T^{2} $
	83	$ 1 - 156.T + 6.88e3T^{2} $
	89	$ 1 + 101. iT - 7.92e3T^{2} $
	97	$ 1 - 41iT - 9.40e3T^{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

−9.781537230348024429378767142071, −9.042157940271478658695075146416, −8.212934610956349717360217601211, −7.24845914529468342622250852168, −6.47072041713408641715902750489, −5.45921994544957157861369148042, −4.48265890245514940491915796223, −3.44649684363635337589631808091, −2.25445787935851589081579259691, −0.76595321914697998349363635259, 1.02536836385813590530597792080, 2.49274221219668928834850719830, 3.52874024704247942383791841188, 4.76820743709061340870524540804, 5.50477521694918878815404360112, 6.66226846612237448348965728284, 7.35880681456355697767175199760, 8.387335286087498593078348555211, 9.094339398579304223684424107069, 10.03092781830903136533306634155

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.840822557\)
\(L(\frac12)\)	\(\approx\)	\(1.840822557\)
\(L(2)\)		not available
\(L(1)\)		not available

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