LMFDB - L-function 2-675-15.14-c4-0-26

Dirichlet series

L(s) = 1

− 16·4^-s + 71i·7^-s + 337i·13^-s + 256·16^-s + 601·19^-s − 1.13e3i·28^-s + 194·31^-s − 529i·37^-s + 3.21e3i·43^-s − 2.64e3·49^-s − 5.39e3i·52^-s + 7.19e3·61^-s − 4.09e3·64^-s + 2.90e3i·67^-s + 1.24e3i·73^-s + ⋯

L(s) = 1

− 4^-s + 1.44i·7^-s + 1.99i·13^-s + 16^-s + 1.66·19^-s − 1.44i·28^-s + 0.201·31^-s − 0.386i·37^-s + 1.73i·43^-s − 1.09·49^-s − 1.99i·52^-s + 1.93·61^-s − 64^-s + 0.646i·67^-s + 0.234i·73^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(5-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$675$ = $3^{3} \cdot 5^{2}$
Sign:	$-0.894 - 0.447i$
Analytic conductor:	$69.7747$
Root analytic conductor:	$8.35312$
Motivic weight:	$4$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{675} (674, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 675,\ (\ :2),\ -0.894 - 0.447i)$

Particular Values

$L(\frac{5}{2})$	$\approx$	$1.270296992$
$L(\frac12)$	$\approx$	$1.270296992$
$L(3)$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	3	$ 1 $
	5	$ 1 $
good	2	$ 1 + 16T^{2} $
	7	$ 1 - 71iT - 2.40e3T^{2} $
	11	$ 1 - 1.46e4T^{2} $
	13	$ 1 - 337iT - 2.85e4T^{2} $
	17	$ 1 + 8.35e4T^{2} $
	19	$ 1 - 601T + 1.30e5T^{2} $
	23	$ 1 + 2.79e5T^{2} $
	29	$ 1 - 7.07e5T^{2} $
	31	$ 1 - 194T + 9.23e5T^{2} $
	37	$ 1 + 529iT - 1.87e6T^{2} $
	41	$ 1 - 2.82e6T^{2} $
	43	$ 1 - 3.21e3iT - 3.41e6T^{2} $
	47	$ 1 + 4.87e6T^{2} $
	53	$ 1 + 7.89e6T^{2} $
	59	$ 1 - 1.21e7T^{2} $
	61	$ 1 - 7.19e3T + 1.38e7T^{2} $
	67	$ 1 - 2.90e3iT - 2.01e7T^{2} $
	71	$ 1 - 2.54e7T^{2} $
	73	$ 1 - 1.24e3iT - 2.83e7T^{2} $
	79	$ 1 + 4.67e3T + 3.89e7T^{2} $
	83	$ 1 + 4.74e7T^{2} $
	89	$ 1 - 6.27e7T^{2} $
	97	$ 1 - 9.07e3iT - 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

−9.894136935261747224246831294055, −9.302554764757078308511571489217, −8.817613746913329213004613426290, −7.84823228163789168840728039079, −6.65735750761638823408619214456, −5.63133227551627336983248135923, −4.88405243157734951181548894172, −3.86373502740978822561609578483, −2.60463092346776473999780761415, −1.31786610418475026016977186174, 0.40340272717512578559109886609, 1.04700143309265243011532730213, 3.14026216801121257144814884402, 3.85253598000753849579066963183, 4.99194969112450031288050188446, 5.67411452038575278954233766722, 7.15821692788371587781678537949, 7.79725461788637088876286471328, 8.581901453981873824212234206977, 9.840932813220387346003300432212

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

\(L(\frac{5}{2})\)	\(\approx\)	\(1.270296992\)
\(L(\frac12)\)	\(\approx\)	\(1.270296992\)
\(L(3)\)		not available
\(L(1)\)		not available

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