LMFDB - L-function 2-1960-5.4-c1-0-12

Dirichlet series

L(s) = 1

− 1.12i·3^-s + (−0.981 − 2.00i)5^-s + 1.74·9^-s − 4.94·11^-s + 4.64i·13^-s + (−2.25 + 1.10i)15^-s + 5.72i·17^-s + 7.53·19^-s + 6.87i·23^-s + (−3.07 + 3.94i)25^-s − 5.31i·27^-s − 8.11·29^-s + 3.87·31^-s + 5.54i·33^-s + 5.18i·37^-s + ⋯

L(s) = 1

− 0.647i·3^-s + (−0.438 − 0.898i)5^-s + 0.580·9^-s − 1.49·11^-s + 1.28i·13^-s + (−0.582 + 0.284i)15^-s + 1.38i·17^-s + 1.72·19^-s + 1.43i·23^-s + (−0.615 + 0.788i)25^-s − 1.02i·27^-s − 1.50·29^-s + 0.696·31^-s + 0.965i·33^-s + 0.852i·37^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1960$ = $2^{3} \cdot 5 \cdot 7^{2}$
Sign:	$0.898 - 0.438i$
Analytic conductor:	$15.6506$
Root analytic conductor:	$3.95609$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1960} (1569, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1960,\ (\ :1/2),\ 0.898 - 0.438i)$

Particular Values

$L(1)$	$\approx$	$1.281213212$
$L(\frac12)$	$\approx$	$1.281213212$
$L(\frac{3}{2})$		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 + (0.981 + 2.00i)T $
	7	$ 1 $
good	3	$ 1 + 1.12iT - 3T^{2} $
	11	$ 1 + 4.94T + 11T^{2} $
	13	$ 1 - 4.64iT - 13T^{2} $
	17	$ 1 - 5.72iT - 17T^{2} $
	19	$ 1 - 7.53T + 19T^{2} $
	23	$ 1 - 6.87iT - 23T^{2} $
	29	$ 1 + 8.11T + 29T^{2} $
	31	$ 1 - 3.87T + 31T^{2} $
	37	$ 1 - 5.18iT - 37T^{2} $
	41	$ 1 - 9.45T + 41T^{2} $
	43	$ 1 - 0.706iT - 43T^{2} $
	47	$ 1 + 2.20iT - 47T^{2} $
	53	$ 1 - 1.32iT - 53T^{2} $
	59	$ 1 + 3.94T + 59T^{2} $
	61	$ 1 - 6.07T + 61T^{2} $
	67	$ 1 + 8.24iT - 67T^{2} $
	71	$ 1 + 4.50T + 71T^{2} $
	73	$ 1 - 2.93iT - 73T^{2} $
	79	$ 1 - 11.0T + 79T^{2} $
	83	$ 1 + 5.27iT - 83T^{2} $
	89	$ 1 + 9.49T + 89T^{2} $
	97	$ 1 - 1.74iT - 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

−9.322746036618155760435207930448, −8.200823465600453089069020629006, −7.69453205543485433768713033554, −7.16261307998597165852261176527, −5.97735662118072135793304518074, −5.24443825272504838813583929614, −4.35262324894234925155094413441, −3.46438826564374813013969386738, −2.01090848560333777006788455974, −1.17585074116199065601782808231, 0.51813025889866790338274529204, 2.60642997009145317582918682730, 3.08872155250847089044016871653, 4.16088624062800184526191474433, 5.13497437062279405092778504275, 5.69078061156451997383623142658, 7.05370390708368208943485532874, 7.53291021458942877978418470832, 8.117979483815698025075127902261, 9.423804059830501639545949402640

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.281213212\)
\(L(\frac12)\)	\(\approx\)	\(1.281213212\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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