LMFDB - L-function 2-1216-152.37-c2-0-0

Dirichlet series

L(s) = 1

− 4.81·3^-s + 9.51i·5^-s + 8.13·7^-s + 14.2·9^-s + 2.56i·11^-s − 18.6·13^-s − 45.8i·15^-s − 17.5·17^-s + (−18.8 + 2.12i)19^-s − 39.1·21^-s + 23.2·23^-s − 65.5·25^-s − 25.0·27^-s + 41.6·29^-s + 15.2i·31^-s + ⋯

L(s) = 1

− 1.60·3^-s + 1.90i·5^-s + 1.16·7^-s + 1.57·9^-s + 0.233i·11^-s − 1.43·13^-s − 3.05i·15^-s − 1.03·17^-s + (−0.993 + 0.111i)19^-s − 1.86·21^-s + 1.00·23^-s − 2.62·25^-s − 0.927·27^-s + 1.43·29^-s + 0.493i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1216$ = $2^{6} \cdot 19$
Sign:	$-0.149 + 0.988i$
Analytic conductor:	$33.1336$
Root analytic conductor:	$5.75617$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1216} (417, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1216,\ (\ :1),\ -0.149 + 0.988i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.04729809803$
$L(\frac12)$	$\approx$	$0.04729809803$
$L(2)$		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 $
	19	$ 1 + (18.8 - 2.12i)T $
good	3	$ 1 + 4.81T + 9T^{2} $
	5	$ 1 - 9.51iT - 25T^{2} $
	7	$ 1 - 8.13T + 49T^{2} $
	11	$ 1 - 2.56iT - 121T^{2} $
	13	$ 1 + 18.6T + 169T^{2} $
	17	$ 1 + 17.5T + 289T^{2} $
	23	$ 1 - 23.2T + 529T^{2} $
	29	$ 1 - 41.6T + 841T^{2} $
	31	$ 1 - 15.2iT - 961T^{2} $
	37	$ 1 + 13.4T + 1.36e3T^{2} $
	41	$ 1 - 74.2iT - 1.68e3T^{2} $
	43	$ 1 + 45.3iT - 1.84e3T^{2} $
	47	$ 1 + 44.0T + 2.20e3T^{2} $
	53	$ 1 + 42.9T + 2.80e3T^{2} $
	59	$ 1 + 23.3T + 3.48e3T^{2} $
	61	$ 1 + 34.5iT - 3.72e3T^{2} $
	67	$ 1 - 36.9T + 4.48e3T^{2} $
	71	$ 1 - 48.8iT - 5.04e3T^{2} $
	73	$ 1 + 67.0T + 5.32e3T^{2} $
	79	$ 1 + 91.4iT - 6.24e3T^{2} $
	83	$ 1 - 37.4iT - 6.88e3T^{2} $
	89	$ 1 + 95.7iT - 7.92e3T^{2} $
	97	$ 1 - 111. iT - 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

−10.46676099314946570396306118297, −9.766912074025639805981062953130, −8.294271704666360719756760980804, −7.25575657758101981536284804422, −6.77573915238550951481891794654, −6.16584211517629125210922159692, −4.95433261373626271131453664111, −4.51383100403130384702420090349, −2.90118481714571345301512597082, −1.84841025669040758396460036505, 0.02107689280514475957285240956, 0.946483987004520937371814761180, 2.03155385843744886902800130740, 4.46849427748959647079313610536, 4.72126418227011671553808012888, 5.25367109030992322002779457954, 6.20996589113865113700984851695, 7.23327128908968721862375450642, 8.254564882344114172131224537658, 8.872277997556400793178654560928

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.04729809803\)
\(L(\frac12)\)	\(\approx\)	\(0.04729809803\)
\(L(2)\)		not available
\(L(1)\)		not available

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