LMFDB - L-function 2-1323-21.20-c1-0-4

Dirichlet series

L(s) = 1

− 2.02i·2^-s − 2.09·4^-s + 0.182·5^-s + 0.197i·8^-s − 0.369i·10^-s + 3.75i·11^-s + 5.85i·13^-s − 3.79·16^-s − 4.31·17^-s + 3.82i·19^-s − 0.383·20^-s + 7.60·22^-s − 4.66i·23^-s − 4.96·25^-s + 11.8·26^-s + ⋯

L(s) = 1

− 1.43i·2^-s − 1.04·4^-s + 0.0817·5^-s + 0.0699i·8^-s − 0.116i·10^-s + 1.13i·11^-s + 1.62i·13^-s − 0.948·16^-s − 1.04·17^-s + 0.877i·19^-s − 0.0857·20^-s + 1.62·22^-s − 0.972i·23^-s − 0.993·25^-s + 2.32·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1323$ = $3^{3} \cdot 7^{2}$
Sign:	$0.912 - 0.409i$
Analytic conductor:	$10.5642$
Root analytic conductor:	$3.25026$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1323} (1322, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1323,\ (\ :1/2),\ 0.912 - 0.409i)$

Particular Values

$L(1)$	$\approx$	$0.8691707657$
$L(\frac12)$	$\approx$	$0.8691707657$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} $

	$p$	$F_p(T)$
bad	3	$ 1 $
	7	$ 1 $
good	2	$ 1 + 2.02iT - 2T^{2} $
	5	$ 1 - 0.182T + 5T^{2} $
	11	$ 1 - 3.75iT - 11T^{2} $
	13	$ 1 - 5.85iT - 13T^{2} $
	17	$ 1 + 4.31T + 17T^{2} $
	19	$ 1 - 3.82iT - 19T^{2} $
	23	$ 1 + 4.66iT - 23T^{2} $
	29	$ 1 - 5.70iT - 29T^{2} $
	31	$ 1 - 8.38iT - 31T^{2} $
	37	$ 1 + 8.44T + 37T^{2} $
	41	$ 1 + 1.51T + 41T^{2} $
	43	$ 1 + 3.00T + 43T^{2} $
	47	$ 1 + 0.591T + 47T^{2} $
	53	$ 1 + 13.0iT - 53T^{2} $
	59	$ 1 - 10.0T + 59T^{2} $
	61	$ 1 + 1.40iT - 61T^{2} $
	67	$ 1 - 7.78T + 67T^{2} $
	71	$ 1 - 5.78iT - 71T^{2} $
	73	$ 1 - 4.53iT - 73T^{2} $
	79	$ 1 + 7.88T + 79T^{2} $
	83	$ 1 + 11.3T + 83T^{2} $
	89	$ 1 - 17.6T + 89T^{2} $
	97	$ 1 - 2.33iT - 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

−9.989976962163632181555446615340, −9.077075870164304377037966621807, −8.500776795015430051334679219652, −6.93329834626335333241247024576, −6.71823113289293323235596780698, −5.06937741307993870019850843393, −4.29809553949309342579521715574, −3.52488133296701001564473010834, −2.14241786212797646603567500943, −1.69134803915280677268001655573, 0.33809296441659317267188177522, 2.41248205139627763938006995710, 3.63986549980872038772997453406, 4.83453848092688392247971203329, 5.74467110268696589494380863828, 6.09772572881351847220576602024, 7.19409628406792305808016867198, 7.86246068603969238024462915859, 8.507642950771994151019783253446, 9.240706663786496291886868444215

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8691707657\)
\(L(\frac12)\)	\(\approx\)	\(0.8691707657\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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