LMFDB - L-function 2-3168-8.5-c1-0-44

Dirichlet series

L(s) = 1

− 2.21i·5^-s − 1.26·7^-s − i·11^-s + 0.807i·13^-s + 5.69·17^-s − 5.87i·19^-s − 7.89·23^-s + 0.109·25^-s − 7.17i·29^-s − 7.31·31^-s + 2.79i·35^-s + 9.62i·37^-s + 2.83·41^-s + 2.83i·43^-s + 7.22·47^-s + ⋯

L(s) = 1

− 0.989i·5^-s − 0.476·7^-s − 0.301i·11^-s + 0.223i·13^-s + 1.38·17^-s − 1.34i·19^-s − 1.64·23^-s + 0.0218·25^-s − 1.33i·29^-s − 1.31·31^-s + 0.471i·35^-s + 1.58i·37^-s + 0.441·41^-s + 0.431i·43^-s + 1.05·47^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3168$ = $2^{5} \cdot 3^{2} \cdot 11$
Sign:	$-0.939 + 0.342i$
Analytic conductor:	$25.2966$
Root analytic conductor:	$5.02957$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3168} (1585, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3168,\ (\ :1/2),\ -0.939 + 0.342i)$

Particular Values

$L(1)$	$\approx$	$0.8914353113$
$L(\frac12)$	$\approx$	$0.8914353113$
$L(\frac{3}{2})$		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 $
	3	$ 1 $
	11	$ 1 + iT $
good	5	$ 1 + 2.21iT - 5T^{2} $
	7	$ 1 + 1.26T + 7T^{2} $
	13	$ 1 - 0.807iT - 13T^{2} $
	17	$ 1 - 5.69T + 17T^{2} $
	19	$ 1 + 5.87iT - 19T^{2} $
	23	$ 1 + 7.89T + 23T^{2} $
	29	$ 1 + 7.17iT - 29T^{2} $
	31	$ 1 + 7.31T + 31T^{2} $
	37	$ 1 - 9.62iT - 37T^{2} $
	41	$ 1 - 2.83T + 41T^{2} $
	43	$ 1 - 2.83iT - 43T^{2} $
	47	$ 1 - 7.22T + 47T^{2} $
	53	$ 1 + 3.01iT - 53T^{2} $
	59	$ 1 - 3.32iT - 59T^{2} $
	61	$ 1 + 8.97iT - 61T^{2} $
	67	$ 1 + 12.3iT - 67T^{2} $
	71	$ 1 - 2.56T + 71T^{2} $
	73	$ 1 + 9.01T + 73T^{2} $
	79	$ 1 + 2.35T + 79T^{2} $
	83	$ 1 - 2.50iT - 83T^{2} $
	89	$ 1 + 0.832T + 89T^{2} $
	97	$ 1 + 15.0T + 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

−8.240026039546206473381226544178, −7.83147470522383643314036490193, −6.78555989283951847813036582851, −6.00240240222987665640692791468, −5.29362935405304286827297673307, −4.47958969458961455297630488038, −3.64619227781912938538097978178, −2.64681204703493806406321575286, −1.41864145836190885568084068143, −0.27652975110918122732431732305, 1.51296397377198029706681033901, 2.61455137008931318729721357871, 3.52018290394455882130532430761, 4.04294255975184050014504147862, 5.60112805292028428340225545106, 5.78419411819108883078189661139, 6.90146773429164425851859655839, 7.43728729206927333751997635490, 8.097306340500195875957524200326, 9.092671731160339041051301415394

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

\(L(1)\)	\(\approx\)	\(0.8914353113\)
\(L(\frac12)\)	\(\approx\)	\(0.8914353113\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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