LMFDB - L-function 2-1792-8.5-c1-0-2

Dirichlet series

L(s) = 1

− 2.73i·3^-s − 0.732i·5^-s − 7^-s − 4.46·9^-s + 5.46i·11^-s + 0.732i·13^-s − 2·15^-s + 0.535·17^-s + 4.19i·19^-s + 2.73i·21^-s − 4.92·23^-s + 4.46·25^-s + 3.99i·27^-s + 3.46i·29^-s − 9.46·31^-s + ⋯

L(s) = 1

− 1.57i·3^-s − 0.327i·5^-s − 0.377·7^-s − 1.48·9^-s + 1.64i·11^-s + 0.203i·13^-s − 0.516·15^-s + 0.129·17^-s + 0.962i·19^-s + 0.596i·21^-s − 1.02·23^-s + 0.892·25^-s + 0.769i·27^-s + 0.643i·29^-s − 1.69·31^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1792$ = $2^{8} \cdot 7$
Sign:	$0.707 - 0.707i$
Analytic conductor:	$14.3091$
Root analytic conductor:	$3.78274$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1792} (897, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1792,\ (\ :1/2),\ 0.707 - 0.707i)$

Particular Values

$L(1)$	$\approx$	$0.8319644540$
$L(\frac12)$	$\approx$	$0.8319644540$
$L(\frac{3}{2})$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 $
	7	$ 1 + T $
good	3	$ 1 + 2.73iT - 3T^{2} $
	5	$ 1 + 0.732iT - 5T^{2} $
	11	$ 1 - 5.46iT - 11T^{2} $
	13	$ 1 - 0.732iT - 13T^{2} $
	17	$ 1 - 0.535T + 17T^{2} $
	19	$ 1 - 4.19iT - 19T^{2} $
	23	$ 1 + 4.92T + 23T^{2} $
	29	$ 1 - 3.46iT - 29T^{2} $
	31	$ 1 + 9.46T + 31T^{2} $
	37	$ 1 - 11.4iT - 37T^{2} $
	41	$ 1 + 3.46T + 41T^{2} $
	43	$ 1 - 9.46iT - 43T^{2} $
	47	$ 1 + 5.46T + 47T^{2} $
	53	$ 1 - 6iT - 53T^{2} $
	59	$ 1 + 4.19iT - 59T^{2} $
	61	$ 1 + 13.1iT - 61T^{2} $
	67	$ 1 + 1.07iT - 67T^{2} $
	71	$ 1 - 13.8T + 71T^{2} $
	73	$ 1 - 4.92T + 73T^{2} $
	79	$ 1 + 8T + 79T^{2} $
	83	$ 1 - 10.7iT - 83T^{2} $
	89	$ 1 - 8.92T + 89T^{2} $
	97	$ 1 - 7.46T + 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

−9.411352443427763204180037788818, −8.245267498995361057312961916834, −7.82145059174890896280433082994, −6.87523836758685490307608159545, −6.54808727925839631043617140681, −5.47088240295501807374238400014, −4.52565990234898475403322559159, −3.26933551527017671160195544551, −2.01734131243831922367429965932, −1.41754123507142383678931044567, 0.30530175464298679185113551224, 2.52734568907067497506701201927, 3.52991646594699326598945976574, 3.91426167913954025207047202909, 5.19717284801658946276285674320, 5.68733741499531846891022655374, 6.64474688854347916418844242186, 7.73429482677527468072889308472, 8.871877055584522219526012957278, 9.023278804875896399871423925372

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(0.8319644540\)
\(L(\frac12)\)	\(\approx\)	\(0.8319644540\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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