LMFDB - L-function 2-960-5.4-c3-0-18

Dirichlet series

L(s) = 1

− 3i·3^-s + (−5 − 10i)5^-s + 4i·7^-s − 9·9^-s + 28·11^-s − 16i·13^-s + (−30 + 15i)15^-s + 108i·17^-s + 32·19^-s + 12·21^-s + 28i·23^-s + (−75 + 100i)25^-s + 27i·27^-s − 238·29^-s − 180·31^-s + ⋯

L(s) = 1

− 0.577i·3^-s + (−0.447 − 0.894i)5^-s + 0.215i·7^-s − 0.333·9^-s + 0.767·11^-s − 0.341i·13^-s + (−0.516 + 0.258i)15^-s + 1.54i·17^-s + 0.386·19^-s + 0.124·21^-s + 0.253i·23^-s + (−0.599 + 0.800i)25^-s + 0.192i·27^-s − 1.52·29^-s − 1.04·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$960$ = $2^{6} \cdot 3 \cdot 5$
Sign:	$0.894 - 0.447i$
Analytic conductor:	$56.6418$
Root analytic conductor:	$7.52607$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{960} (769, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 960,\ (\ :3/2),\ 0.894 - 0.447i)$

Particular Values

$L(2)$	$\approx$	$1.410541120$
$L(\frac12)$	$\approx$	$1.410541120$
$L(\frac{5}{2})$		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 + 3iT $
	5	$ 1 + (5 + 10i)T $
good	7	$ 1 - 4iT - 343T^{2} $
	11	$ 1 - 28T + 1.33e3T^{2} $
	13	$ 1 + 16iT - 2.19e3T^{2} $
	17	$ 1 - 108iT - 4.91e3T^{2} $
	19	$ 1 - 32T + 6.85e3T^{2} $
	23	$ 1 - 28iT - 1.21e4T^{2} $
	29	$ 1 + 238T + 2.43e4T^{2} $
	31	$ 1 + 180T + 2.97e4T^{2} $
	37	$ 1 - 40iT - 5.06e4T^{2} $
	41	$ 1 - 422T + 6.89e4T^{2} $
	43	$ 1 - 276iT - 7.95e4T^{2} $
	47	$ 1 - 60iT - 1.03e5T^{2} $
	53	$ 1 - 220iT - 1.48e5T^{2} $
	59	$ 1 + 804T + 2.05e5T^{2} $
	61	$ 1 - 358T + 2.26e5T^{2} $
	67	$ 1 - 884iT - 3.00e5T^{2} $
	71	$ 1 + 64T + 3.57e5T^{2} $
	73	$ 1 - 152iT - 3.89e5T^{2} $
	79	$ 1 - 932T + 4.93e5T^{2} $
	83	$ 1 + 1.29e3iT - 5.71e5T^{2} $
	89	$ 1 - 1.14e3T + 7.04e5T^{2} $
	97	$ 1 - 824iT - 9.12e5T^{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

−9.375669761276239025467676266918, −8.921617590435870099819280121792, −7.940231604187553626058958271784, −7.40581940253367705944549159150, −6.13658793907514267403701130081, −5.54396947160107077586116768728, −4.29680459460580734505408145264, −3.47767673272146598574600325041, −1.90944381370871878544777922199, −0.993314128233941416643496697739, 0.42363763931967096986831234987, 2.20491403632821746159877990550, 3.38026130916138417331494975619, 4.05114616346553013144725261851, 5.13972110641544922493447694840, 6.20349161443271520272515510734, 7.17358872072716866533678592072, 7.66939592366608689363062534726, 9.088790357346323675094785603966, 9.437970580805497472742977447913

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.410541120\)
\(L(\frac12)\)	\(\approx\)	\(1.410541120\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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