LMFDB - L-function 2-8280-69.68-c1-0-28

Dirichlet series

L(s) = 1

− 5^-s − 4.29i·7^-s + 0.813·11^-s + 1.00·13^-s + 7.09·17^-s + 3.82i·19^-s + (−1.12 + 4.66i)23^-s + 25^-s + 0.328i·29^-s − 4.61·31^-s + 4.29i·35^-s + 10.4i·37^-s − 3.42i·41^-s + 7.16i·43^-s − 0.556i·47^-s + ⋯

L(s) = 1

− 0.447·5^-s − 1.62i·7^-s + 0.245·11^-s + 0.277·13^-s + 1.72·17^-s + 0.877i·19^-s + (−0.234 + 0.972i)23^-s + 0.200·25^-s + 0.0610i·29^-s − 0.827·31^-s + 0.725i·35^-s + 1.71i·37^-s − 0.534i·41^-s + 1.09i·43^-s − 0.0812i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$8280$ = $2^{3} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$0.752 - 0.658i$
Analytic conductor:	$66.1161$
Root analytic conductor:	$8.13118$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{8280} (1241, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 8280,\ (\ :1/2),\ 0.752 - 0.658i)$

Particular Values

$L(1)$	$\approx$	$1.624387148$
$L(\frac12)$	$\approx$	$1.624387148$
$L(\frac{3}{2})$		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 $
	5	$ 1 + T $
	23	$ 1 + (1.12 - 4.66i)T $
good	7	$ 1 + 4.29iT - 7T^{2} $
	11	$ 1 - 0.813T + 11T^{2} $
	13	$ 1 - 1.00T + 13T^{2} $
	17	$ 1 - 7.09T + 17T^{2} $
	19	$ 1 - 3.82iT - 19T^{2} $
	29	$ 1 - 0.328iT - 29T^{2} $
	31	$ 1 + 4.61T + 31T^{2} $
	37	$ 1 - 10.4iT - 37T^{2} $
	41	$ 1 + 3.42iT - 41T^{2} $
	43	$ 1 - 7.16iT - 43T^{2} $
	47	$ 1 + 0.556iT - 47T^{2} $
	53	$ 1 - 3.55T + 53T^{2} $
	59	$ 1 - 12.4iT - 59T^{2} $
	61	$ 1 + 9.65iT - 61T^{2} $
	67	$ 1 - 1.86iT - 67T^{2} $
	71	$ 1 + 0.163iT - 71T^{2} $
	73	$ 1 + 8.45T + 73T^{2} $
	79	$ 1 - 15.2iT - 79T^{2} $
	83	$ 1 - 5.16T + 83T^{2} $
	89	$ 1 - 3.52T + 89T^{2} $
	97	$ 1 - 7.94iT - 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

−7.81143547702699971336303073394, −7.33589020447524234300789116445, −6.64095297650430627388404478333, −5.80641549171482265156770705612, −5.08248951211694982160414002741, −4.11911652558111268024778349574, −3.68832753616811325926738114940, −3.07398952628643599558501505894, −1.50405666678383903507911093736, −0.986884422587548469929893999234, 0.44761598166255615340275840626, 1.75850489527580064704112567893, 2.60861886154868677900726879603, 3.32105430865216574638371151061, 4.13824718292637272090918456274, 5.12014622690909898611409951055, 5.61227938576786429172557476886, 6.22786483938895281428862423834, 7.09482244365266241380952318261, 7.78449741888068680769188247288

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.624387148\)
\(L(\frac12)\)	\(\approx\)	\(1.624387148\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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