LMFDB - L-function 2-72-3.2-c2-0-1

Dirichlet series

L(s) = 1

− 7.07i·5^-s + 12·7^-s + 5.65i·11^-s − 8·13^-s − 9.89i·17^-s − 16·19^-s + 39.5i·23^-s − 25.0·25^-s + 29.6i·29^-s − 4·31^-s − 84.8i·35^-s + 30·37^-s + 21.2i·41^-s − 8·43^-s + 16.9i·47^-s + ⋯

L(s) = 1

− 1.41i·5^-s + 1.71·7^-s + 0.514i·11^-s − 0.615·13^-s − 0.582i·17^-s − 0.842·19^-s + 1.72i·23^-s − 1.00·25^-s + 1.02i·29^-s − 0.129·31^-s − 2.42i·35^-s + 0.810·37^-s + 0.517i·41^-s − 0.186·43^-s + 0.361i·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$72$ = $2^{3} \cdot 3^{2}$
Sign:	$0.816 + 0.577i$
Analytic conductor:	$1.96185$
Root analytic conductor:	$1.40066$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{72} (17, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 72,\ (\ :1),\ 0.816 + 0.577i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.28757 - 0.409238i$
$L(\frac12)$	$\approx$	$1.28757 - 0.409238i$
$L(2)$		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 $
good	5	$ 1 + 7.07iT - 25T^{2} $
	7	$ 1 - 12T + 49T^{2} $
	11	$ 1 - 5.65iT - 121T^{2} $
	13	$ 1 + 8T + 169T^{2} $
	17	$ 1 + 9.89iT - 289T^{2} $
	19	$ 1 + 16T + 361T^{2} $
	23	$ 1 - 39.5iT - 529T^{2} $
	29	$ 1 - 29.6iT - 841T^{2} $
	31	$ 1 + 4T + 961T^{2} $
	37	$ 1 - 30T + 1.36e3T^{2} $
	41	$ 1 - 21.2iT - 1.68e3T^{2} $
	43	$ 1 + 8T + 1.84e3T^{2} $
	47	$ 1 - 16.9iT - 2.20e3T^{2} $
	53	$ 1 + 49.4iT - 2.80e3T^{2} $
	59	$ 1 + 79.1iT - 3.48e3T^{2} $
	61	$ 1 + 14T + 3.72e3T^{2} $
	67	$ 1 + 88T + 4.48e3T^{2} $
	71	$ 1 - 28.2iT - 5.04e3T^{2} $
	73	$ 1 + 80T + 5.32e3T^{2} $
	79	$ 1 - 100T + 6.24e3T^{2} $
	83	$ 1 - 130. iT - 6.88e3T^{2} $
	89	$ 1 + 148. iT - 7.92e3T^{2} $
	97	$ 1 + 112T + 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

−14.33688358682227208617919456638, −13.11125846431616065470282926631, −12.08726861334488215259499615832, −11.17347642642200155994255737107, −9.562931416427124915648568021834, −8.479495284912885758686002255949, −7.50482026346804560290118092431, −5.28151627139828152533807794972, −4.54446825434988345631970049446, −1.60717280542503331801809039300, 2.40195417189720467656724289281, 4.41047057046745592853859020617, 6.14129096528961839903305573819, 7.50046611581818985907348929049, 8.540730622267136602260144436666, 10.43259805922958889231493771656, 10.98837051696760615676023155450, 12.07479092609245996145900417107, 13.72844626951643900372702828138, 14.84427364537398920610985588318

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.28757 - 0.409238i\)
\(L(\frac12)\)	\(\approx\)	\(1.28757 - 0.409238i\)
\(L(2)\)		not available
\(L(1)\)		not available

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