LMFDB - L-function 2-2e10-8.5-c3-0-72

Dirichlet series

L(s) = 1

− 1.07i·3^-s + 11.6i·5^-s + 2.67·7^-s + 25.8·9^-s − 63.9i·11^-s − 50.0i·13^-s + 12.4·15^-s + 72.4·17^-s + 27.4i·19^-s − 2.85i·21^-s − 139.·23^-s − 10.3·25^-s − 56.5i·27^-s + 93.3i·29^-s − 188.·31^-s + ⋯

L(s) = 1

− 0.205i·3^-s + 1.04i·5^-s + 0.144·7^-s + 0.957·9^-s − 1.75i·11^-s − 1.06i·13^-s + 0.214·15^-s + 1.03·17^-s + 0.332i·19^-s − 0.0297i·21^-s − 1.26·23^-s − 0.0826·25^-s − 0.403i·27^-s + 0.598i·29^-s − 1.09·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1024$ = $2^{10}$
Sign:	$i$
Analytic conductor:	$60.4179$
Root analytic conductor:	$7.77289$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1024} (513, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1024,\ (\ :3/2),\ i)$

Particular Values

$L(2)$	$\approx$	$1.727766883$
$L(\frac12)$	$\approx$	$1.727766883$
$L(\frac{5}{2})$		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 $
good	3	$ 1 + 1.07iT - 27T^{2} $
	5	$ 1 - 11.6iT - 125T^{2} $
	7	$ 1 - 2.67T + 343T^{2} $
	11	$ 1 + 63.9iT - 1.33e3T^{2} $
	13	$ 1 + 50.0iT - 2.19e3T^{2} $
	17	$ 1 - 72.4T + 4.91e3T^{2} $
	19	$ 1 - 27.4iT - 6.85e3T^{2} $
	23	$ 1 + 139.T + 1.21e4T^{2} $
	29	$ 1 - 93.3iT - 2.43e4T^{2} $
	31	$ 1 + 188.T + 2.97e4T^{2} $
	37	$ 1 + 118. iT - 5.06e4T^{2} $
	41	$ 1 + 104.T + 6.89e4T^{2} $
	43	$ 1 + 44.5iT - 7.95e4T^{2} $
	47	$ 1 + 488.T + 1.03e5T^{2} $
	53	$ 1 + 211. iT - 1.48e5T^{2} $
	59	$ 1 + 402. iT - 2.05e5T^{2} $
	61	$ 1 + 322. iT - 2.26e5T^{2} $
	67	$ 1 + 196. iT - 3.00e5T^{2} $
	71	$ 1 - 453.T + 3.57e5T^{2} $
	73	$ 1 - 259.T + 3.89e5T^{2} $
	79	$ 1 - 323.T + 4.93e5T^{2} $
	83	$ 1 + 797. iT - 5.71e5T^{2} $
	89	$ 1 - 866.T + 7.04e5T^{2} $
	97	$ 1 + 936.T + 9.12e5T^{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

−9.519975768623303627175592306825, −8.206892430213144372373535734034, −7.82197590558746291025758518326, −6.78490782853473769522827544620, −6.01964212813464593607904323934, −5.19899655542969075539398622359, −3.60219565301878950634830348388, −3.21024796772076960535820625180, −1.74733608780499775319022653878, −0.43573069393050493713773799589, 1.31235818781141497303487840830, 2.03949103415622623766217241381, 3.84802929255001388143033960710, 4.57068707171562021871722964467, 5.14048617427916447225367808129, 6.47593285981272325196326229059, 7.33992164047772251726021385976, 8.054171950078921126514166541377, 9.173130696438043461749094485874, 9.714461910524542795523078845721

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(1.727766883\)
\(L(\frac12)\)	\(\approx\)	\(1.727766883\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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