LMFDB - L-function 2-1152-24.5-c4-0-6

Dirichlet series

L(s) = 1

− 43.8·5^-s + 238i·13^-s − 111. i·17^-s + 1.29e3·25^-s + 1.12e3·29^-s + 1.68e3i·37^-s − 3.16e3i·41^-s − 2.40e3·49^-s + 5.31e3·53^-s + 2.64e3i·61^-s − 1.04e4i·65^-s + 1.05e4·73^-s + 4.89e3i·85^-s + 1.57e4i·89^-s − 1.87e4·97^-s + ⋯

L(s) = 1

− 1.75·5^-s + 1.40i·13^-s − 0.386i·17^-s + 2.07·25^-s + 1.34·29^-s + 1.22i·37^-s − 1.88i·41^-s − 49^-s + 1.89·53^-s + 0.709i·61^-s − 2.46i·65^-s + 1.98·73^-s + 0.677i·85^-s + 1.98i·89^-s − 1.98·97^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(5-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1152$ = $2^{7} \cdot 3^{2}$
Sign:	$-0.985 - 0.169i$
Analytic conductor:	$119.082$
Root analytic conductor:	$10.9124$
Motivic weight:	$4$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1152} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1152,\ (\ :2),\ -0.985 - 0.169i)$

Particular Values

$L(\frac{5}{2})$	$\approx$	$0.3538691961$
$L(\frac12)$	$\approx$	$0.3538691961$
$L(3)$		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 $
	3	$ 1 $
good	5	$ 1 + 43.8T + 625T^{2} $
	7	$ 1 + 2.40e3T^{2} $
	11	$ 1 + 1.46e4T^{2} $
	13	$ 1 - 238iT - 2.85e4T^{2} $
	17	$ 1 + 111. iT - 8.35e4T^{2} $
	19	$ 1 - 1.30e5T^{2} $
	23	$ 1 - 2.79e5T^{2} $
	29	$ 1 - 1.12e3T + 7.07e5T^{2} $
	31	$ 1 + 9.23e5T^{2} $
	37	$ 1 - 1.68e3iT - 1.87e6T^{2} $
	41	$ 1 + 3.16e3iT - 2.82e6T^{2} $
	43	$ 1 - 3.41e6T^{2} $
	47	$ 1 - 4.87e6T^{2} $
	53	$ 1 - 5.31e3T + 7.89e6T^{2} $
	59	$ 1 + 1.21e7T^{2} $
	61	$ 1 - 2.64e3iT - 1.38e7T^{2} $
	67	$ 1 - 2.01e7T^{2} $
	71	$ 1 - 2.54e7T^{2} $
	73	$ 1 - 1.05e4T + 2.83e7T^{2} $
	79	$ 1 + 3.89e7T^{2} $
	83	$ 1 + 4.74e7T^{2} $
	89	$ 1 - 1.57e4iT - 6.27e7T^{2} $
	97	$ 1 + 1.87e4T + 8.85e7T^{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

−9.518554870954146719319263813270, −8.628317497835971983899557935516, −8.076769725062912651324143052018, −7.07809383770952737324027106417, −6.66075198014413905152179670084, −5.16012271771329061693568973682, −4.30216956220706161239601758286, −3.69473416248439531812493915410, −2.54318766127922846877473978730, −1.04141066028886556959907427846, 0.10237640529779271402195796737, 0.990332650397682746876906820316, 2.76109493493058694139961458588, 3.55958149660570249200866395435, 4.40117607063412088941936224306, 5.31038513283702210577144103362, 6.47092876856752099201233419165, 7.39200500761098460435510144464, 8.117883939327408123880364887083, 8.473582296978988089374742020767

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 $\F_{q}$
Belyi maps

\(L(\frac{5}{2})\)	\(\approx\)	\(0.3538691961\)
\(L(\frac12)\)	\(\approx\)	\(0.3538691961\)
\(L(3)\)		not available
\(L(1)\)		not available

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