LMFDB - L-function 2-783-87.86-c0-0-1

Dirichlet series

L(s) = 1

− 0.347·2^-s − 0.879·4^-s + 0.347·7^-s + 0.652·8^-s − 1.53·11^-s + 1.53·13^-s − 0.120·14^-s + 0.652·16^-s + 1.87·17^-s + 0.532·22^-s + 25^-s − 0.532·26^-s − 0.305·28^-s − 29^-s − 0.879·32^-s − 0.652·34^-s + 41^-s + 1.34·44^-s + 1.87·47^-s − 0.879·49^-s − 0.347·50^-s − 1.34·52^-s + 0.226·56^-s + 0.347·58^-s − 0.347·64^-s − 1.87·67^-s − 1.65·68^-s + ⋯

L(s) = 1

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$783$ = $3^{3} \cdot 29$
Sign:	$1$
Analytic conductor:	$0.390767$
Root analytic conductor:	$0.625114$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{783} (782, \cdot )$
Primitive:	yes
Self-dual:	yes
Analytic rank:	$0$
Selberg data:	$(2,\ 783,\ (\ :0),\ 1)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.7215807945$
$L(\frac12)$	$\approx$	$0.7215807945$
$L(1)$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	29	$ 1 + T $
good	2	$ 1 + 0.347T + T^{2} $
	5	$ 1 - T^{2} $
	7	$ 1 - 0.347T + T^{2} $
	11	$ 1 + 1.53T + T^{2} $
	13	$ 1 - 1.53T + T^{2} $
	17	$ 1 - 1.87T + T^{2} $
	19	$ 1 - T^{2} $
	23	$ 1 - T^{2} $
	31	$ 1 - T^{2} $
	37	$ 1 - T^{2} $
	41	$ 1 - T + T^{2} $
	43	$ 1 - T^{2} $
	47	$ 1 - 1.87T + T^{2} $
	53	$ 1 - T^{2} $
	59	$ 1 - T^{2} $
	61	$ 1 - T^{2} $
	67	$ 1 + 1.87T + T^{2} $
	71	$ 1 - T^{2} $
	73	$ 1 - T^{2} $
	79	$ 1 - T^{2} $
	83	$ 1 - T^{2} $
	89	$ 1 + 1.53T + T^{2} $
	97	$ 1 - T^{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

−10.54246163295777669749684612194, −9.629950041970534913569198171816, −8.734894616587127232260737300374, −8.017031260255429137846727603460, −7.43927119451755162368170634009, −5.81111035021783967658538256772, −5.28101135715859422394691132078, −4.11252499666238415612896202582, −3.04710235328819663014331000612, −1.22701418896861080350007978609, 1.22701418896861080350007978609, 3.04710235328819663014331000612, 4.11252499666238415612896202582, 5.28101135715859422394691132078, 5.81111035021783967658538256772, 7.43927119451755162368170634009, 8.017031260255429137846727603460, 8.734894616587127232260737300374, 9.629950041970534913569198171816, 10.54246163295777669749684612194

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 $\F_{q}$
Belyi maps

\(L(\frac{1}{2})\)	\(\approx\)	\(0.7215807945\)
\(L(\frac12)\)	\(\approx\)	\(0.7215807945\)
\(L(1)\)		not available
\(L(1)\)		not available

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