LMFDB - L-function 2-21e2-21.20-c3-0-20

Dirichlet series

L(s) = 1

− 0.641i·2^-s + 7.58·4^-s + 12.0·5^-s − 9.99i·8^-s − 7.73i·10^-s + 43.9i·11^-s + 66.9i·13^-s + 54.3·16^-s + 39.4·17^-s + 57.9i·19^-s + 91.6·20^-s + 28.1·22^-s + 46.1i·23^-s + 20.7·25^-s + 42.9·26^-s + ⋯

L(s) = 1

− 0.226i·2^-s + 0.948·4^-s + 1.07·5^-s − 0.441i·8^-s − 0.244i·10^-s + 1.20i·11^-s + 1.42i·13^-s + 0.848·16^-s + 0.562·17^-s + 0.699i·19^-s + 1.02·20^-s + 0.272·22^-s + 0.418i·23^-s + 0.166·25^-s + 0.323·26^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(4-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.896 - 0.442i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$441$ = $3^{2} \cdot 7^{2}$
Sign:	$0.896 - 0.442i$
Analytic conductor:	$26.0198$
Root analytic conductor:	$5.10096$
Motivic weight:	$3$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{441} (440, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 441,\ (\ :3/2),\ 0.896 - 0.442i)$

Particular Values

$L(2)$	$\approx$	$3.008161471$
$L(\frac12)$	$\approx$	$3.008161471$
$L(\frac{5}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	7	$ 1 $
good	2	$ 1 + 0.641iT - 8T^{2} $
	5	$ 1 - 12.0T + 125T^{2} $
	11	$ 1 - 43.9iT - 1.33e3T^{2} $
	13	$ 1 - 66.9iT - 2.19e3T^{2} $
	17	$ 1 - 39.4T + 4.91e3T^{2} $
	19	$ 1 - 57.9iT - 6.85e3T^{2} $
	23	$ 1 - 46.1iT - 1.21e4T^{2} $
	29	$ 1 + 201. iT - 2.43e4T^{2} $
	31	$ 1 - 185. iT - 2.97e4T^{2} $
	37	$ 1 + 410.T + 5.06e4T^{2} $
	41	$ 1 - 408.T + 6.89e4T^{2} $
	43	$ 1 - 129.T + 7.95e4T^{2} $
	47	$ 1 + 514.T + 1.03e5T^{2} $
	53	$ 1 + 437. iT - 1.48e5T^{2} $
	59	$ 1 - 607.T + 2.05e5T^{2} $
	61	$ 1 + 24.5iT - 2.26e5T^{2} $
	67	$ 1 - 441.T + 3.00e5T^{2} $
	71	$ 1 + 1.15e3iT - 3.57e5T^{2} $
	73	$ 1 + 525. iT - 3.89e5T^{2} $
	79	$ 1 + 151.T + 4.93e5T^{2} $
	83	$ 1 - 1.30e3T + 5.71e5T^{2} $
	89	$ 1 - 583.T + 7.04e5T^{2} $
	97	$ 1 - 1.09e3iT - 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

−10.65686389121502328994845844779, −9.878189295853298934258279593131, −9.342326753600428089425096689166, −7.87567817907360194711930164092, −6.89152618794372112191868831599, −6.22372680808444788452126019303, −5.10325815917408569770984343974, −3.71061129782374049970699993096, −2.17656441778379260711895981922, −1.63240216106412314960360580708, 0.985259640996823624542412597381, 2.43288762692923588319155389515, 3.34081652550077834803519855145, 5.37730165131364728518433466917, 5.81099838338433530376513807238, 6.79138990671732483756598117749, 7.85475043550257560347440580347, 8.733120435129236724787860341816, 9.908326218818108983463789363899, 10.66656964939705459140578059926

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 $\F_{q}$

\(L(2)\)	\(\approx\)	\(3.008161471\)
\(L(\frac12)\)	\(\approx\)	\(3.008161471\)
\(L(\frac{5}{2})\)		not available
\(L(1)\)		not available

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