LMFDB - L-function 2-2160-15.14-c2-0-66

Dirichlet series

L(s) = 1

+ (4 − 3i)5^-s − 6i·7^-s + 21i·11^-s − 15i·13^-s + 23·17^-s − 14·19^-s + 7·23^-s + (7 − 24i)25^-s − 3i·29^-s + 25·31^-s + (−18 − 24i)35^-s + 54i·37^-s − 24i·41^-s − 15i·43^-s + 49·47^-s + ⋯

L(s) = 1

+ (0.800 − 0.600i)5^-s − 0.857i·7^-s + 1.90i·11^-s − 1.15i·13^-s + 1.35·17^-s − 0.736·19^-s + 0.304·23^-s + (0.280 − 0.959i)25^-s − 0.103i·29^-s + 0.806·31^-s + (−0.514 − 0.685i)35^-s + 1.45i·37^-s − 0.585i·41^-s − 0.348i·43^-s + 1.04·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.599 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2160$ = $2^{4} \cdot 3^{3} \cdot 5$
Sign:	$0.599 + 0.799i$
Analytic conductor:	$58.8557$
Root analytic conductor:	$7.67174$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2160} (1889, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2160,\ (\ :1),\ 0.599 + 0.799i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$2.531209806$
$L(\frac12)$	$\approx$	$2.531209806$
$L(2)$		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 + (-4 + 3i)T $
good	7	$ 1 + 6iT - 49T^{2} $
	11	$ 1 - 21iT - 121T^{2} $
	13	$ 1 + 15iT - 169T^{2} $
	17	$ 1 - 23T + 289T^{2} $
	19	$ 1 + 14T + 361T^{2} $
	23	$ 1 - 7T + 529T^{2} $
	29	$ 1 + 3iT - 841T^{2} $
	31	$ 1 - 25T + 961T^{2} $
	37	$ 1 - 54iT - 1.36e3T^{2} $
	41	$ 1 + 24iT - 1.68e3T^{2} $
	43	$ 1 + 15iT - 1.84e3T^{2} $
	47	$ 1 - 49T + 2.20e3T^{2} $
	53	$ 1 - 14T + 2.80e3T^{2} $
	59	$ 1 + 30iT - 3.48e3T^{2} $
	61	$ 1 - 44T + 3.72e3T^{2} $
	67	$ 1 + 66iT - 4.48e3T^{2} $
	71	$ 1 - 18iT - 5.04e3T^{2} $
	73	$ 1 - 5.32e3T^{2} $
	79	$ 1 - 37T + 6.24e3T^{2} $
	83	$ 1 + 116T + 6.88e3T^{2} $
	89	$ 1 + 126iT - 7.92e3T^{2} $
	97	$ 1 - 78iT - 9.40e3T^{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

−8.788992994663698616355236674522, −7.930803258645152067740069378333, −7.29470958958619771238313617964, −6.46575682133373419199520583446, −5.47543344023529242908169832353, −4.81434126231532467440711517544, −4.02181385679837111083763660890, −2.78015530362325746881189242271, −1.70433687238204907052358108119, −0.73857156632404016968590844807, 1.03800421838750038478354674208, 2.28922265167572238473264429065, 3.03131882103692379178825404630, 3.97941813767163814965820891214, 5.35126404017434479731769543747, 5.88107980852728526465272388992, 6.44355303341868154947900732554, 7.40643734124517462809278039862, 8.481322111909778583747951604464, 8.928788149191546541089377415239

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(2.531209806\)
\(L(\frac12)\)	\(\approx\)	\(2.531209806\)
\(L(2)\)		not available
\(L(1)\)		not available

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