LMFDB - L-function 2-3168-44.43-c1-0-41

Dirichlet series

L(s) = 1

− 1.86·5^-s + 3.91·7^-s + (3.30 − 0.269i)11^-s − 3.53i·13^-s − 1.77i·17^-s + 3.02·19^-s + 7.41i·23^-s − 1.50·25^-s − 5.55i·29^-s − 9.85i·31^-s − 7.32·35^-s + 9.85·37^-s + 8.38i·41^-s − 2.26·43^-s + 3.13i·47^-s + ⋯

L(s) = 1

− 0.836·5^-s + 1.48·7^-s + (0.996 − 0.0813i)11^-s − 0.980i·13^-s − 0.429i·17^-s + 0.694·19^-s + 1.54i·23^-s − 0.300·25^-s − 1.03i·29^-s − 1.76i·31^-s − 1.23·35^-s + 1.62·37^-s + 1.30i·41^-s − 0.345·43^-s + 0.456i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3168$ = $2^{5} \cdot 3^{2} \cdot 11$
Sign:	$0.762 + 0.647i$
Analytic conductor:	$25.2966$
Root analytic conductor:	$5.02957$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3168} (703, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3168,\ (\ :1/2),\ 0.762 + 0.647i)$

Particular Values

$L(1)$	$\approx$	$2.005219527$
$L(\frac12)$	$\approx$	$2.005219527$
$L(\frac{3}{2})$		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 $
	11	$ 1 + (-3.30 + 0.269i)T $
good	5	$ 1 + 1.86T + 5T^{2} $
	7	$ 1 - 3.91T + 7T^{2} $
	13	$ 1 + 3.53iT - 13T^{2} $
	17	$ 1 + 1.77iT - 17T^{2} $
	19	$ 1 - 3.02T + 19T^{2} $
	23	$ 1 - 7.41iT - 23T^{2} $
	29	$ 1 + 5.55iT - 29T^{2} $
	31	$ 1 + 9.85iT - 31T^{2} $
	37	$ 1 - 9.85T + 37T^{2} $
	41	$ 1 - 8.38iT - 41T^{2} $
	43	$ 1 + 2.26T + 43T^{2} $
	47	$ 1 - 3.13iT - 47T^{2} $
	53	$ 1 + 8.13T + 53T^{2} $
	59	$ 1 + 10.5iT - 59T^{2} $
	61	$ 1 + 8.82iT - 61T^{2} $
	67	$ 1 + 4.84iT - 67T^{2} $
	71	$ 1 - 0.607iT - 71T^{2} $
	73	$ 1 - 13.8iT - 73T^{2} $
	79	$ 1 + 3.15T + 79T^{2} $
	83	$ 1 - 6.37T + 83T^{2} $
	89	$ 1 + 15.9T + 89T^{2} $
	97	$ 1 - 9.34T + 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

−8.227080748424149228816906404499, −7.87584111094597633039372179067, −7.48182888417748494482584660989, −6.25153102112154930740022135641, −5.49946663743303762436535294097, −4.62625408326843188053845777664, −3.98399586179645150977347442664, −3.06629861735314785612471765593, −1.78823534438991200898584245996, −0.75433269867036962013398481923, 1.12347621292684529418623564779, 1.97508278213573388080110008019, 3.30861883301933583140866181703, 4.33240787224152817041795406732, 4.58727988162399793591738599159, 5.66348831904745439863575430831, 6.71344637996032918960852856646, 7.26509933053507258707203957887, 8.092445127386432068838319138669, 8.692217222326032466396264582002

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(2.005219527\)
\(L(\frac12)\)	\(\approx\)	\(2.005219527\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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