LMFDB - L-function 2-9900-5.4-c1-0-27

Dirichlet series

L(s) = 1

− 0.208i·7^-s + 11^-s + i·13^-s − 0.791i·17^-s − 6.58·19^-s − 3.79i·23^-s + 6.79·29^-s − 8.58·31^-s − 2.58i·37^-s + 1.41·41^-s + 10i·43^-s + 1.41i·47^-s + 6.95·49^-s + 11.3i·53^-s − 10.5·59^-s + ⋯

L(s) = 1

− 0.0788i·7^-s + 0.301·11^-s + 0.277i·13^-s − 0.191i·17^-s − 1.51·19^-s − 0.790i·23^-s + 1.26·29^-s − 1.54·31^-s − 0.424i·37^-s + 0.221·41^-s + 1.52i·43^-s + 0.206i·47^-s + 0.993·49^-s + 1.56i·53^-s − 1.37·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 9900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$9900$ = $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$79.0518$
Root analytic conductor:	$8.89111$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{9900} (5149, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 9900,\ (\ :1/2),\ 0.894 - 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.717557030$
$L(\frac12)$	$\approx$	$1.717557030$
$L(\frac{3}{2})$		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 $
	11	$ 1 - T $
good	7	$ 1 + 0.208iT - 7T^{2} $
	13	$ 1 - iT - 13T^{2} $
	17	$ 1 + 0.791iT - 17T^{2} $
	19	$ 1 + 6.58T + 19T^{2} $
	23	$ 1 + 3.79iT - 23T^{2} $
	29	$ 1 - 6.79T + 29T^{2} $
	31	$ 1 + 8.58T + 31T^{2} $
	37	$ 1 + 2.58iT - 37T^{2} $
	41	$ 1 - 1.41T + 41T^{2} $
	43	$ 1 - 10iT - 43T^{2} $
	47	$ 1 - 1.41iT - 47T^{2} $
	53	$ 1 - 11.3iT - 53T^{2} $
	59	$ 1 + 10.5T + 59T^{2} $
	61	$ 1 - 4.20T + 61T^{2} $
	67	$ 1 + 4iT - 67T^{2} $
	71	$ 1 - 10.7T + 71T^{2} $
	73	$ 1 - 7.79iT - 73T^{2} $
	79	$ 1 - 15.5T + 79T^{2} $
	83	$ 1 + 9.95iT - 83T^{2} $
	89	$ 1 + 0.791T + 89T^{2} $
	97	$ 1 + 6.20iT - 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

−7.68225068098159930220986036940, −7.00361081116260628424845911656, −6.31863844172517422154891497080, −5.85947672904456259620413236446, −4.73901995725784631866277136249, −4.37710401466911563980523490988, −3.51232768243304598262609002093, −2.59346482903948967936327661477, −1.85107277238180022418646870874, −0.71961071004049629909059080833, 0.52579458505651217309570551696, 1.74671569369418801167875883130, 2.42109471644520601309162377921, 3.52233628927630057796817808312, 3.98533275039838462210827855752, 4.95012824678705333446608635528, 5.51244837548085123247581326151, 6.38856062403074853073185290391, 6.80050801330763321914707965780, 7.67311512148211317287848508788

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.717557030\)
\(L(\frac12)\)	\(\approx\)	\(1.717557030\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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