LMFDB - L-function 2-1323-21.20-c1-0-3

Dirichlet series

L(s) = 1

− 1.10i·2^-s + 0.786·4^-s − 2.47·5^-s − 3.06i·8^-s + 2.72i·10^-s + 3.81i·11^-s + 0.163i·13^-s − 1.80·16^-s − 5.85·17^-s + 7.11i·19^-s − 1.94·20^-s + 4.20·22^-s + 5.54i·23^-s + 1.13·25^-s + 0.180·26^-s + ⋯

L(s) = 1

− 0.779i·2^-s + 0.393·4^-s − 1.10·5^-s − 1.08i·8^-s + 0.862i·10^-s + 1.14i·11^-s + 0.0453i·13^-s − 0.452·16^-s − 1.41·17^-s + 1.63i·19^-s − 0.435·20^-s + 0.895·22^-s + 1.15i·23^-s + 0.226·25^-s + 0.0353·26^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$1323$ = $3^{3} \cdot 7^{2}$
Sign:	$0.409 - 0.912i$
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:	$(2,\ 1323,\ (\ :1/2),\ 0.409 - 0.912i)$

Particular Values

$L(1)$	$\approx$	$0.7891506200$
$L(\frac12)$	$\approx$	$0.7891506200$
$L(\frac{3}{2})$		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 + 1.10iT - 2T^{2} $
	5	$ 1 + 2.47T + 5T^{2} $
	11	$ 1 - 3.81iT - 11T^{2} $
	13	$ 1 - 0.163iT - 13T^{2} $
	17	$ 1 + 5.85T + 17T^{2} $
	19	$ 1 - 7.11iT - 19T^{2} $
	23	$ 1 - 5.54iT - 23T^{2} $
	29	$ 1 + 3.32iT - 29T^{2} $
	31	$ 1 + 0.133iT - 31T^{2} $
	37	$ 1 + 7.98T + 37T^{2} $
	41	$ 1 + 9.65T + 41T^{2} $
	43	$ 1 - 12.0T + 43T^{2} $
	47	$ 1 - 5.79T + 47T^{2} $
	53	$ 1 - 8.50iT - 53T^{2} $
	59	$ 1 - 3.09T + 59T^{2} $
	61	$ 1 - 3.22iT - 61T^{2} $
	67	$ 1 - 7.59T + 67T^{2} $
	71	$ 1 - 12.8iT - 71T^{2} $
	73	$ 1 - 10.5iT - 73T^{2} $
	79	$ 1 + 10.3T + 79T^{2} $
	83	$ 1 - 7.08T + 83T^{2} $
	89	$ 1 - 4.95T + 89T^{2} $
	97	$ 1 - 4.99iT - 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

−9.988605352053489959143654027383, −9.133145085620113788867831620830, −8.061873580480997101096418984794, −7.34143344664754894797859491991, −6.73109161307025347441491014019, −5.54796065747128744292454578362, −4.13818436584347781232862214305, −3.85677345296596725182232914855, −2.52308840401338491750643387224, −1.53645524094314341949886932546, 0.31000501906190090338490452643, 2.32838634252152242015743309214, 3.35600250812551567449941629779, 4.49999914074365699364375296274, 5.33376675550522198227347036630, 6.51001467303605894640250499640, 6.89932004483300122240844263293, 7.79979972469636124944137682409, 8.629735393518724340652308804732, 8.942152837439480935688019382223

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.7891506200\)
\(L(\frac12)\)	\(\approx\)	\(0.7891506200\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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