LMFDB - L-function 2-675-15.14-c2-0-6

Dirichlet series

L(s) = 1

− 1.25·2^-s − 2.43·4^-s + 7.62i·7^-s + 8.05·8^-s + 15.8i·11^-s − 10.7i·13^-s − 9.54i·14^-s − 0.352·16^-s + 1.55·17^-s + 23.9·19^-s − 19.7i·22^-s − 33.5·23^-s + 13.3i·26^-s − 18.5i·28^-s − 22.0i·29^-s + ⋯

L(s) = 1

− 0.626·2^-s − 0.608·4^-s + 1.08i·7^-s + 1.00·8^-s + 1.43i·11^-s − 0.823i·13^-s − 0.681i·14^-s − 0.0220·16^-s + 0.0916·17^-s + 1.26·19^-s − 0.899i·22^-s − 1.45·23^-s + 0.515i·26^-s − 0.662i·28^-s − 0.760i·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]

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

Invariants

Degree:	$2$
Conductor:	$675$ = $3^{3} \cdot 5^{2}$
Sign:	$-0.894 - 0.447i$
Analytic conductor:	$18.3924$
Root analytic conductor:	$4.28863$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{675} (674, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 675,\ (\ :1),\ -0.894 - 0.447i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.5426573894$
$L(\frac12)$	$\approx$	$0.5426573894$
$L(2)$		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 + 1.25T + 4T^{2} $
	7	$ 1 - 7.62iT - 49T^{2} $
	11	$ 1 - 15.8iT - 121T^{2} $
	13	$ 1 + 10.7iT - 169T^{2} $
	17	$ 1 - 1.55T + 289T^{2} $
	19	$ 1 - 23.9T + 361T^{2} $
	23	$ 1 + 33.5T + 529T^{2} $
	29	$ 1 + 22.0iT - 841T^{2} $
	31	$ 1 - 27.0T + 961T^{2} $
	37	$ 1 - 28.7iT - 1.36e3T^{2} $
	41	$ 1 - 48.2iT - 1.68e3T^{2} $
	43	$ 1 + 9.02iT - 1.84e3T^{2} $
	47	$ 1 + 48.8T + 2.20e3T^{2} $
	53	$ 1 + 59.6T + 2.80e3T^{2} $
	59	$ 1 - 99.8iT - 3.48e3T^{2} $
	61	$ 1 + 89.2T + 3.72e3T^{2} $
	67	$ 1 - 1.85iT - 4.48e3T^{2} $
	71	$ 1 - 126. iT - 5.04e3T^{2} $
	73	$ 1 + 101. iT - 5.32e3T^{2} $
	79	$ 1 + 108.T + 6.24e3T^{2} $
	83	$ 1 + 3.54T + 6.88e3T^{2} $
	89	$ 1 + 108. iT - 7.92e3T^{2} $
	97	$ 1 + 88.8iT - 9.40e3T^{2} $
show more
show less

$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$

Imaginary part of the first few zeros on the critical line

−10.08183289679515253733512319666, −9.934472303233586905059191602657, −9.021257237368028049315098381344, −8.062833090409821445954092678668, −7.55486481917330388432629028979, −6.15929772573683284827724798436, −5.17622863667633939765805250601, −4.34048492030556916759982748081, −2.84255686687177084424978280592, −1.49719110251746132132684143156, 0.27299783750831176187415578193, 1.41817343711524251144352439399, 3.39071391377200142011140146360, 4.22108690657305320322250587361, 5.33335275293745735185829642483, 6.49215518649692812400009596000, 7.57115618524398238528393110150, 8.192543154895431798433316406768, 9.129113913866660388093120379358, 9.838110560700434992202809463594

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

\(L(\frac{3}{2})\)	\(\approx\)	\(0.5426573894\)
\(L(\frac12)\)	\(\approx\)	\(0.5426573894\)
\(L(2)\)		not available
\(L(1)\)		not available

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Origins

Related objects

Downloads

Learn more