LMFDB - L-function 2-1216-152.75-c1-0-18

Dirichlet series

L(s) = 1

− 2.27i·3^-s + 4.53i·7^-s − 2.16·9^-s − 6.13·13^-s + 8.23·17^-s − 4.35i·19^-s + 10.3·21^-s − 2.86i·23^-s + 5·25^-s − 1.90i·27^-s + 10.6·29^-s + 8.71·37^-s + 13.9i·39^-s − 6i·47^-s − 13.5·49^-s + ⋯

L(s) = 1

− 1.31i·3^-s + 1.71i·7^-s − 0.721·9^-s − 1.70·13^-s + 1.99·17^-s − 0.999i·19^-s + 2.24·21^-s − 0.596i·23^-s + 25^-s − 0.365i·27^-s + 1.98·29^-s + 1.43·37^-s + 2.23i·39^-s − 0.875i·47^-s − 1.93·49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1216$ = $2^{6} \cdot 19$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$9.70980$
Root analytic conductor:	$3.11605$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1216} (607, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1216,\ (\ :1/2),\ 0.707 + 0.707i)$

Particular Values

$L(1)$	$\approx$	$1.646915236$
$L(\frac12)$	$\approx$	$1.646915236$
$L(\frac{3}{2})$		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 $
	19	$ 1 + 4.35iT $
good	3	$ 1 + 2.27iT - 3T^{2} $
	5	$ 1 - 5T^{2} $
	7	$ 1 - 4.53iT - 7T^{2} $
	11	$ 1 + 11T^{2} $
	13	$ 1 + 6.13T + 13T^{2} $
	17	$ 1 - 8.23T + 17T^{2} $
	23	$ 1 + 2.86iT - 23T^{2} $
	29	$ 1 - 10.6T + 29T^{2} $
	31	$ 1 + 31T^{2} $
	37	$ 1 - 8.71T + 37T^{2} $
	41	$ 1 - 41T^{2} $
	43	$ 1 + 43T^{2} $
	47	$ 1 + 6iT - 47T^{2} $
	53	$ 1 - 2.95T + 53T^{2} $
	59	$ 1 - 14.5iT - 59T^{2} $
	61	$ 1 - 61T^{2} $
	67	$ 1 - 5.44iT - 67T^{2} $
	71	$ 1 + 71T^{2} $
	73	$ 1 - 15.6T + 73T^{2} $
	79	$ 1 + 79T^{2} $
	83	$ 1 + 83T^{2} $
	89	$ 1 - 89T^{2} $
	97	$ 1 - 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

−9.550657184560204757759654269107, −8.637423423320998626023004055252, −7.960405021139775983946064144449, −7.15732572974214134239064446544, −6.40339508809444720153832924106, −5.49276929665662341063713062184, −4.77682314229197190075553929930, −2.73827443079808372802454640087, −2.51054089492660453812627288728, −0.963458208018100116838881629643, 1.02530149836196833590649164211, 3.01055252533244251913754319744, 3.79995782820614073504658218266, 4.65565593395405416654858457734, 5.22757840029787960855859941329, 6.56444340437325956396901072867, 7.56858719772299876678717381202, 8.009119422180190036800312882315, 9.504087893590616468756249506970, 9.954129244305278410782361784373

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.646915236\)
\(L(\frac12)\)	\(\approx\)	\(1.646915236\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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