LMFDB - L-function 2-224-4.3-c2-0-8

Dirichlet series

L(s) = 1

− 2.55i·3^-s + 9.86·5^-s − 2.64i·7^-s + 2.47·9^-s + 13.1i·11^-s − 5.86·13^-s − 25.2i·15^-s − 0.570·17^-s − 15.6i·19^-s − 6.75·21^-s + 16.4i·23^-s + 72.3·25^-s − 29.3i·27^-s − 29.7·29^-s − 54.8i·31^-s + ⋯

L(s) = 1

− 0.851i·3^-s + 1.97·5^-s − 0.377i·7^-s + 0.274·9^-s + 1.19i·11^-s − 0.451·13^-s − 1.68i·15^-s − 0.0335·17^-s − 0.824i·19^-s − 0.321·21^-s + 0.716i·23^-s + 2.89·25^-s − 1.08i·27^-s − 1.02·29^-s − 1.76i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$224$ = $2^{5} \cdot 7$
Sign:	$0.707 + 0.707i$
Analytic conductor:	$6.10355$
Root analytic conductor:	$2.47053$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{224} (127, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 224,\ (\ :1),\ 0.707 + 0.707i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.97408 - 0.817693i$
$L(\frac12)$	$\approx$	$1.97408 - 0.817693i$
$L(2)$		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 $
	7	$ 1 + 2.64iT $
good	3	$ 1 + 2.55iT - 9T^{2} $
	5	$ 1 - 9.86T + 25T^{2} $
	11	$ 1 - 13.1iT - 121T^{2} $
	13	$ 1 + 5.86T + 169T^{2} $
	17	$ 1 + 0.570T + 289T^{2} $
	19	$ 1 + 15.6iT - 361T^{2} $
	23	$ 1 - 16.4iT - 529T^{2} $
	29	$ 1 + 29.7T + 841T^{2} $
	31	$ 1 + 54.8iT - 961T^{2} $
	37	$ 1 + 42.0T + 1.36e3T^{2} $
	41	$ 1 - 0.773T + 1.68e3T^{2} $
	43	$ 1 - 41.7iT - 1.84e3T^{2} $
	47	$ 1 - 58.4iT - 2.20e3T^{2} $
	53	$ 1 - 5.65T + 2.80e3T^{2} $
	59	$ 1 - 42.6iT - 3.48e3T^{2} $
	61	$ 1 + 95.9T + 3.72e3T^{2} $
	67	$ 1 - 69.8iT - 4.48e3T^{2} $
	71	$ 1 + 92.0iT - 5.04e3T^{2} $
	73	$ 1 - 9.97T + 5.32e3T^{2} $
	79	$ 1 + 20.1iT - 6.24e3T^{2} $
	83	$ 1 - 151. iT - 6.88e3T^{2} $
	89	$ 1 - 5.79T + 7.92e3T^{2} $
	97	$ 1 - 103.T + 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

−12.21126460813623934465187386659, −10.78926818907266438136548721512, −9.739204899573004597039945102407, −9.357300196076271496049307469757, −7.58887242433735179077140025045, −6.83676097967662332829249063265, −5.85835182976470628779714902298, −4.66008621705535079831659915351, −2.40456534071269030521606587998, −1.50098197212991132308958801179, 1.79837948263342960039456953692, 3.29861522692108960745638425725, 5.03800970056913530524778985981, 5.73278127810613671751350318584, 6.80420481128837437157313154216, 8.657437091751355184823086594959, 9.305445802290685809528723453070, 10.28971681132868143795494725762, 10.66995356125795292875394202670, 12.26164882891466705626303154017

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(1.97408 - 0.817693i\)
\(L(\frac12)\)	\(\approx\)	\(1.97408 - 0.817693i\)
\(L(2)\)		not available
\(L(1)\)		not available

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