LMFDB - L-function 2-675-15.14-c2-0-31

Dirichlet series

L(s) = 1

− 3·2^-s + 5·4^-s + 5i·7^-s − 3·8^-s − 15i·11^-s + 10i·13^-s − 15i·14^-s − 11·16^-s − 18·17^-s + 16·19^-s + 45i·22^-s − 12·23^-s − 30i·26^-s + 25i·28^-s − 30i·29^-s + ⋯

L(s) = 1

− 1.5·2^-s + 1.25·4^-s + 0.714i·7^-s − 0.375·8^-s − 1.36i·11^-s + 0.769i·13^-s − 1.07i·14^-s − 0.687·16^-s − 1.05·17^-s + 0.842·19^-s + 2.04i·22^-s − 0.521·23^-s − 1.15i·26^-s + 0.892i·28^-s − 1.03i·29^-s + ⋯

Functional equation

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

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

Invariants

Degree:	$2$
Conductor:	$675$ = $3^{3} \cdot 5^{2}$
Sign:	$-0.447 + 0.894i$
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:	$(2,\ 675,\ (\ :1),\ -0.447 + 0.894i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.3436486048$
$L(\frac12)$	$\approx$	$0.3436486048$
$L(2)$		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 + 3T + 4T^{2} $
	7	$ 1 - 5iT - 49T^{2} $
	11	$ 1 + 15iT - 121T^{2} $
	13	$ 1 - 10iT - 169T^{2} $
	17	$ 1 + 18T + 289T^{2} $
	19	$ 1 - 16T + 361T^{2} $
	23	$ 1 + 12T + 529T^{2} $
	29	$ 1 + 30iT - 841T^{2} $
	31	$ 1 + T + 961T^{2} $
	37	$ 1 - 20iT - 1.36e3T^{2} $
	41	$ 1 - 60iT - 1.68e3T^{2} $
	43	$ 1 + 50iT - 1.84e3T^{2} $
	47	$ 1 - 6T + 2.20e3T^{2} $
	53	$ 1 + 27T + 2.80e3T^{2} $
	59	$ 1 - 30iT - 3.48e3T^{2} $
	61	$ 1 + 76T + 3.72e3T^{2} $
	67	$ 1 + 10iT - 4.48e3T^{2} $
	71	$ 1 + 90iT - 5.04e3T^{2} $
	73	$ 1 + 65iT - 5.32e3T^{2} $
	79	$ 1 + 14T + 6.24e3T^{2} $
	83	$ 1 - 3T + 6.88e3T^{2} $
	89	$ 1 + 90iT - 7.92e3T^{2} $
	97	$ 1 + 85iT - 9.40e3T^{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

−9.764660104936171163957372035636, −9.074454212027712804358714379561, −8.503681465233164055953518857584, −7.71706047600381822482973033001, −6.62940193708529391490958579519, −5.83106377365469666439550108981, −4.43013711287894528641938817387, −2.89456883386571471674014301072, −1.69397909185269059751482143819, −0.22688459197100730241639971440, 1.18841066289640365644732696884, 2.39970783882283772074681650816, 4.02672067894159593110209727028, 5.12884436492597590519634348419, 6.64608105608474697997117176960, 7.34309539424457261980046934253, 7.938451530493115918041309698093, 9.009479974122409186242934937186, 9.650335263566465326960880338029, 10.44404718309194642428073701084

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.3436486048\)
\(L(\frac12)\)	\(\approx\)	\(0.3436486048\)
\(L(2)\)		not available
\(L(1)\)		not available

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