LMFDB - L-function 2-1900-5.4-c1-0-15

Dirichlet series

L(s) = 1

− 0.286i·3^-s + 0.286i·7^-s + 2.91·9^-s + 4.26·11^-s − 3.20i·13^-s + 0.286i·17^-s + 19^-s + 0.0820·21^-s − 0.936i·23^-s − 1.69i·27^-s − 2.26·29^-s − 4.18·31^-s − 1.22i·33^-s + 8.67i·37^-s − 0.917·39^-s + ⋯

L(s) = 1

− 0.165i·3^-s + 0.108i·7^-s + 0.972·9^-s + 1.28·11^-s − 0.888i·13^-s + 0.0694i·17^-s + 0.229·19^-s + 0.0179·21^-s − 0.195i·23^-s − 0.326i·27^-s − 0.421·29^-s − 0.751·31^-s − 0.212i·33^-s + 1.42i·37^-s − 0.146·39^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1900$ = $2^{2} \cdot 5^{2} \cdot 19$
Sign:	$0.894 + 0.447i$
Analytic conductor:	$15.1715$
Root analytic conductor:	$3.89507$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1900} (1749, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1900,\ (\ :1/2),\ 0.894 + 0.447i)$

Particular Values

$L(1)$	$\approx$	$2.053653533$
$L(\frac12)$	$\approx$	$2.053653533$
$L(\frac{3}{2})$		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 $
	5	$ 1 $
	19	$ 1 - T $
good	3	$ 1 + 0.286iT - 3T^{2} $
	7	$ 1 - 0.286iT - 7T^{2} $
	11	$ 1 - 4.26T + 11T^{2} $
	13	$ 1 + 3.20iT - 13T^{2} $
	17	$ 1 - 0.286iT - 17T^{2} $
	23	$ 1 + 0.936iT - 23T^{2} $
	29	$ 1 + 2.26T + 29T^{2} $
	31	$ 1 + 4.18T + 31T^{2} $
	37	$ 1 - 8.67iT - 37T^{2} $
	41	$ 1 - 1.08T + 41T^{2} $
	43	$ 1 + 4.42iT - 43T^{2} $
	47	$ 1 + 0.759iT - 47T^{2} $
	53	$ 1 + 4.42iT - 53T^{2} $
	59	$ 1 - 4.70T + 59T^{2} $
	61	$ 1 - 10.1T + 61T^{2} $
	67	$ 1 - 9.82iT - 67T^{2} $
	71	$ 1 - 4.83T + 71T^{2} $
	73	$ 1 + 10.4iT - 73T^{2} $
	79	$ 1 + 5.10T + 79T^{2} $
	83	$ 1 + 15.7iT - 83T^{2} $
	89	$ 1 - 9.75T + 89T^{2} $
	97	$ 1 + 9.61iT - 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

−9.147468625404384210512509117878, −8.417690652998357140592606852738, −7.47599283759351528889419058937, −6.86471737345315234014819316682, −6.04356803875479508894617692000, −5.10969867883336478248094966466, −4.12133811345971645441563824884, −3.37006498760838785035399721635, −2.01635699316324936724975159383, −0.945721141435967371517106898310, 1.15942764922985240014791125707, 2.17164383396088408427077589074, 3.75214478955797858849550791251, 4.10565635326503537390454307865, 5.16710890266188848408535406145, 6.19780950302329501767272831110, 6.99044297766891802743986379254, 7.49436806210320418128641937317, 8.684254841581479412463612592128, 9.374274621990568686553775057224

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.053653533\)
\(L(\frac12)\)	\(\approx\)	\(2.053653533\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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