LMFDB - L-function 2-3900-65.64-c1-0-18

Dirichlet series

L(s) = 1

− i·3^-s + 4·7^-s − 9^-s + 6i·11^-s + (2 − 3i)13^-s − 2i·17^-s − 4i·21^-s + 8i·23^-s + i·27^-s − 2·29^-s + 8i·31^-s + 6·33^-s + 8·37^-s + (−3 − 2i)39^-s + 2i·41^-s + ⋯

L(s) = 1

− 0.577i·3^-s + 1.51·7^-s − 0.333·9^-s + 1.80i·11^-s + (0.554 − 0.832i)13^-s − 0.485i·17^-s − 0.872i·21^-s + 1.66i·23^-s + 0.192i·27^-s − 0.371·29^-s + 1.43i·31^-s + 1.04·33^-s + 1.31·37^-s + (−0.480 − 0.320i)39^-s + 0.312i·41^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3900$ = $2^{2} \cdot 3 \cdot 5^{2} \cdot 13$
Sign:	$0.868 - 0.496i$
Analytic conductor:	$31.1416$
Root analytic conductor:	$5.58047$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3900} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3900,\ (\ :1/2),\ 0.868 - 0.496i)$

Particular Values

$L(1)$	$\approx$	$2.237492406$
$L(\frac12)$	$\approx$	$2.237492406$
$L(\frac{3}{2})$		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 + iT $
	5	$ 1 $
	13	$ 1 + (-2 + 3i)T $
good	7	$ 1 - 4T + 7T^{2} $
	11	$ 1 - 6iT - 11T^{2} $
	17	$ 1 + 2iT - 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 - 8iT - 23T^{2} $
	29	$ 1 + 2T + 29T^{2} $
	31	$ 1 - 8iT - 31T^{2} $
	37	$ 1 - 8T + 37T^{2} $
	41	$ 1 - 2iT - 41T^{2} $
	43	$ 1 - 8iT - 43T^{2} $
	47	$ 1 + 6T + 47T^{2} $
	53	$ 1 + 6iT - 53T^{2} $
	59	$ 1 + 2iT - 59T^{2} $
	61	$ 1 - 2T + 61T^{2} $
	67	$ 1 - 4T + 67T^{2} $
	71	$ 1 + 6iT - 71T^{2} $
	73	$ 1 - 4T + 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 14T + 83T^{2} $
	89	$ 1 - 6iT - 89T^{2} $
	97	$ 1 + 12T + 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

−8.212270599074477458945089243701, −7.82191525244218656714120657842, −7.25464747561921049960145300029, −6.44911292266521089479928789523, −5.30181484594887880187252715995, −4.99111881040837292500799548405, −4.03146824389988410141666798673, −2.88299206858567516784440152848, −1.81120982090884251691259084557, −1.26272626178530119458712496451, 0.69690918146664954369575445452, 1.89769973023955191234965786354, 2.92199222402403104648715948344, 4.09277081562035586842513033470, 4.37695134297838192755396296187, 5.54168721813336039769789088529, 5.95006416716229798043281073834, 6.89853985219508709986383893001, 8.080577411629150387887007090149, 8.365480525752042951043630283436

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.237492406\)
\(L(\frac12)\)	\(\approx\)	\(2.237492406\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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