LMFDB - L-function 2-828-3.2-c2-0-15

Dirichlet series

L(s) = 1

− 9.02i·5^-s − 2.91·7^-s + 14.2i·11^-s − 12.0·13^-s − 29.9i·17^-s + 0.454·19^-s + 4.79i·23^-s − 56.5·25^-s − 11.9i·29^-s − 2.91·31^-s + 26.3i·35^-s − 8.71·37^-s + 48.3i·41^-s − 7.20·43^-s + 86.6i·47^-s + ⋯

L(s) = 1

− 1.80i·5^-s − 0.416·7^-s + 1.29i·11^-s − 0.930·13^-s − 1.76i·17^-s + 0.0239·19^-s + 0.208i·23^-s − 2.26·25^-s − 0.413i·29^-s − 0.0941·31^-s + 0.752i·35^-s − 0.235·37^-s + 1.18i·41^-s − 0.167·43^-s + 1.84i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$828$ = $2^{2} \cdot 3^{2} \cdot 23$
Sign:	$-0.816 - 0.577i$
Analytic conductor:	$22.5613$
Root analytic conductor:	$4.74988$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{828} (737, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 828,\ (\ :1),\ -0.816 - 0.577i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.2271662210$
$L(\frac12)$	$\approx$	$0.2271662210$
$L(2)$		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 $
	23	$ 1 - 4.79iT $
good	5	$ 1 + 9.02iT - 25T^{2} $
	7	$ 1 + 2.91T + 49T^{2} $
	11	$ 1 - 14.2iT - 121T^{2} $
	13	$ 1 + 12.0T + 169T^{2} $
	17	$ 1 + 29.9iT - 289T^{2} $
	19	$ 1 - 0.454T + 361T^{2} $
	29	$ 1 + 11.9iT - 841T^{2} $
	31	$ 1 + 2.91T + 961T^{2} $
	37	$ 1 + 8.71T + 1.36e3T^{2} $
	41	$ 1 - 48.3iT - 1.68e3T^{2} $
	43	$ 1 + 7.20T + 1.84e3T^{2} $
	47	$ 1 - 86.6iT - 2.20e3T^{2} $
	53	$ 1 - 52.9iT - 2.80e3T^{2} $
	59	$ 1 - 7.37iT - 3.48e3T^{2} $
	61	$ 1 + 0.539T + 3.72e3T^{2} $
	67	$ 1 + 29.5T + 4.48e3T^{2} $
	71	$ 1 - 79.0iT - 5.04e3T^{2} $
	73	$ 1 + 24.9T + 5.32e3T^{2} $
	79	$ 1 + 147.T + 6.24e3T^{2} $
	83	$ 1 + 111. iT - 6.88e3T^{2} $
	89	$ 1 + 42.8iT - 7.92e3T^{2} $
	97	$ 1 - 59.7T + 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

−9.660311690576274003255109420981, −8.849701062485596779776894477424, −7.76294669194065507748680884989, −7.13102989528913180104062113254, −5.78578844225775765834944147297, −4.75305860561401216816989822328, −4.50556403080178279487770620450, −2.76171957746924006547280543152, −1.42482384678850932977240771424, −0.07396563123345468736963978273, 2.10864495798692065023750931409, 3.19629240555094966374555596383, 3.80391112358663877722353596227, 5.49651690501909150338809066175, 6.34354436424423869647444297359, 6.92838708985239732816594006719, 7.895777690601270824547440215317, 8.784834776720293369932085842588, 10.02461530797751789870607479069, 10.47939456940140400096921030986

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

\(L(\frac{3}{2})\)	\(\approx\)	\(0.2271662210\)
\(L(\frac12)\)	\(\approx\)	\(0.2271662210\)
\(L(2)\)		not available
\(L(1)\)		not available

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