LMFDB - L-function 2-2816-88.43-c1-0-76

Dirichlet series

L(s) = 1

− 3.31·3^-s − 3i·5^-s + 8·9^-s − 3.31·11^-s + 9.94i·15^-s − 3.31i·23^-s − 4·25^-s − 16.5·27^-s − 9.94i·31^-s + 11·33^-s − 7i·37^-s − 24i·45^-s + 6.63i·47^-s − 7·49^-s − 6i·53^-s + ⋯

L(s) = 1

− 1.91·3^-s − 1.34i·5^-s + 2.66·9^-s − 1.00·11^-s + 2.56i·15^-s − 0.691i·23^-s − 0.800·25^-s − 3.19·27^-s − 1.78i·31^-s + 1.91·33^-s − 1.15i·37^-s − 3.57i·45^-s + 0.967i·47^-s − 49^-s − 0.824i·53^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2816 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2816$ = $2^{8} \cdot 11$
Sign:	$-0.707 - 0.707i$
Analytic conductor:	$22.4858$
Root analytic conductor:	$4.74192$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2816} (1407, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2816,\ (\ :1/2),\ -0.707 - 0.707i)$

Particular Values

$L(1)$	$\approx$	$0.2087358338$
$L(\frac12)$	$\approx$	$0.2087358338$
$L(\frac{3}{2})$		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 $
	11	$ 1 + 3.31T $
good	3	$ 1 + 3.31T + 3T^{2} $
	5	$ 1 + 3iT - 5T^{2} $
	7	$ 1 + 7T^{2} $
	13	$ 1 + 13T^{2} $
	17	$ 1 - 17T^{2} $
	19	$ 1 - 19T^{2} $
	23	$ 1 + 3.31iT - 23T^{2} $
	29	$ 1 + 29T^{2} $
	31	$ 1 + 9.94iT - 31T^{2} $
	37	$ 1 + 7iT - 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 - 43T^{2} $
	47	$ 1 - 6.63iT - 47T^{2} $
	53	$ 1 + 6iT - 53T^{2} $
	59	$ 1 - 3.31T + 59T^{2} $
	61	$ 1 + 61T^{2} $
	67	$ 1 - 9.94T + 67T^{2} $
	71	$ 1 + 16.5iT - 71T^{2} $
	73	$ 1 - 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 - 83T^{2} $
	89	$ 1 - 9T + 89T^{2} $
	97	$ 1 + 17T + 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

−8.158228127808683227837063593392, −7.56551561111861001077255308239, −6.54100661110838616488795690122, −5.85770081209442332159804974212, −5.19291021906610373757256407671, −4.72575608159544033466697369664, −3.94794735948926606505467989861, −2.12515996259660670686997468995, −0.904865739246197683215103865476, −0.11684236313958310069999527090, 1.41724734698628641674531263699, 2.76887423399141173296664278410, 3.79919458488347069675570406478, 4.93836560042813342069750601640, 5.41939771897886739534084915664, 6.27495287544410493997397845685, 6.88223902451638137998332737987, 7.32967725164141542307095565570, 8.325936412173401365142047803675, 9.762592720623508386208167690654

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

\(L(1)\)	\(\approx\)	\(0.2087358338\)
\(L(\frac12)\)	\(\approx\)	\(0.2087358338\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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