LMFDB - L-function 2-176-44.43-c5-0-29

Dirichlet series

L(s) = 1

− 26.5i·3^-s − 36.9·5^-s + 239.·7^-s − 463.·9^-s + (−334. − 221. i)11^-s − 809. i·13^-s + 981. i·15^-s − 32.3i·17^-s − 1.38e3·19^-s − 6.37e3i·21^-s − 2.85e3i·23^-s − 1.76e3·25^-s + 5.86e3i·27^-s + 5.50e3i·29^-s + 3.47e3i·31^-s + ⋯

L(s) = 1

− 1.70i·3^-s − 0.660·5^-s + 1.85·7^-s − 1.90·9^-s + (−0.834 − 0.551i)11^-s − 1.32i·13^-s + 1.12i·15^-s − 0.0271i·17^-s − 0.882·19^-s − 3.15i·21^-s − 1.12i·23^-s − 0.563·25^-s + 1.54i·27^-s + 1.21i·29^-s + 0.650i·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(6-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$176$ = $2^{4} \cdot 11$
Sign:	$-0.834 - 0.551i$
Analytic conductor:	$28.2275$
Root analytic conductor:	$5.31296$
Motivic weight:	$5$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{176} (175, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 176,\ (\ :5/2),\ -0.834 - 0.551i)$

Particular Values

$L(3)$	$\approx$	$1.101725069$
$L(\frac12)$	$\approx$	$1.101725069$
$L(\frac{7}{2})$		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 $
	11	$ 1 + (334. + 221. i)T $
good	3	$ 1 + 26.5iT - 243T^{2} $
	5	$ 1 + 36.9T + 3.12e3T^{2} $
	7	$ 1 - 239.T + 1.68e4T^{2} $
	13	$ 1 + 809. iT - 3.71e5T^{2} $
	17	$ 1 + 32.3iT - 1.41e6T^{2} $
	19	$ 1 + 1.38e3T + 2.47e6T^{2} $
	23	$ 1 + 2.85e3iT - 6.43e6T^{2} $
	29	$ 1 - 5.50e3iT - 2.05e7T^{2} $
	31	$ 1 - 3.47e3iT - 2.86e7T^{2} $
	37	$ 1 - 4.53e3T + 6.93e7T^{2} $
	41	$ 1 - 5.08e3iT - 1.15e8T^{2} $
	43	$ 1 - 4.92e3T + 1.47e8T^{2} $
	47	$ 1 - 2.29e4iT - 2.29e8T^{2} $
	53	$ 1 + 2.91e4T + 4.18e8T^{2} $
	59	$ 1 + 2.19e4iT - 7.14e8T^{2} $
	61	$ 1 + 4.58e4iT - 8.44e8T^{2} $
	67	$ 1 - 3.44e4iT - 1.35e9T^{2} $
	71	$ 1 - 4.48e4iT - 1.80e9T^{2} $
	73	$ 1 + 7.78e4iT - 2.07e9T^{2} $
	79	$ 1 + 3.65e4T + 3.07e9T^{2} $
	83	$ 1 + 8.73e4T + 3.93e9T^{2} $
	89	$ 1 - 3.23e4T + 5.58e9T^{2} $
	97	$ 1 + 5.64e3T + 8.58e9T^{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

−11.25092766470513776955844243682, −10.77667363786424637878241742354, −8.332299085563701470500004290952, −8.173401587359646823939500531801, −7.34471824310460511418831974523, −5.95787616937708341107192528055, −4.81036657012217212531437740192, −2.79669057717730306235889548124, −1.51830898195249978888536874360, −0.34770020171799485587626065041, 2.11742681877453322650657300623, 4.07965047917725690850330917900, 4.49445134775857577540973376391, 5.55119998612726417047264064492, 7.57465773464946439799922231242, 8.424609167784472321516794241052, 9.461917974429044934909071378724, 10.50138967514065633632960139976, 11.36453984169423526257358379034, 11.78840516819466987829211805468

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 $\F_{q}$

\(L(3)\)	\(\approx\)	\(1.101725069\)
\(L(\frac12)\)	\(\approx\)	\(1.101725069\)
\(L(\frac{7}{2})\)		not available
\(L(1)\)		not available

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