LMFDB - L-function 2-3150-15.14-c2-0-30

Dirichlet series

L(s) = 1

− 1.41·2^-s + 2.00·4^-s − 2.64i·7^-s − 2.82·8^-s + 17.7i·11^-s + 2.58i·13^-s + 3.74i·14^-s + 4.00·16^-s − 25.8·17^-s − 20·19^-s − 25.1i·22^-s − 17.7·23^-s − 3.65i·26^-s − 5.29i·28^-s − 11.9i·29^-s + ⋯

L(s) = 1

− 0.707·2^-s + 0.500·4^-s − 0.377i·7^-s − 0.353·8^-s + 1.61i·11^-s + 0.198i·13^-s + 0.267i·14^-s + 0.250·16^-s − 1.52·17^-s − 1.05·19^-s − 1.14i·22^-s − 0.773·23^-s − 0.140i·26^-s − 0.188i·28^-s − 0.410i·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$3150$ = $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$
Sign:	$0.881 + 0.472i$
Analytic conductor:	$85.8312$
Root analytic conductor:	$9.26451$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3150} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 3150,\ (\ :1),\ 0.881 + 0.472i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$0.8674480170$
$L(\frac12)$	$\approx$	$0.8674480170$
$L(2)$		not available
$L(1)$		not available

Euler product

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

	$p$	$F_p(T)$
bad	2	$ 1 + 1.41T $
	3	$ 1 $
	5	$ 1 $
	7	$ 1 + 2.64iT $
good	11	$ 1 - 17.7iT - 121T^{2} $
	13	$ 1 - 2.58iT - 169T^{2} $
	17	$ 1 + 25.8T + 289T^{2} $
	19	$ 1 + 20T + 361T^{2} $
	23	$ 1 + 17.7T + 529T^{2} $
	29	$ 1 + 11.9iT - 841T^{2} $
	31	$ 1 + 17.1T + 961T^{2} $
	37	$ 1 + 38iT - 1.36e3T^{2} $
	41	$ 1 + 15.7iT - 1.68e3T^{2} $
	43	$ 1 + 43.4iT - 1.84e3T^{2} $
	47	$ 1 + 16.9T + 2.20e3T^{2} $
	53	$ 1 - 85.5T + 2.80e3T^{2} $
	59	$ 1 + 1.64iT - 3.48e3T^{2} $
	61	$ 1 - 100.T + 3.72e3T^{2} $
	67	$ 1 + 36.6iT - 4.48e3T^{2} $
	71	$ 1 - 17.7iT - 5.04e3T^{2} $
	73	$ 1 - 28.9iT - 5.32e3T^{2} $
	79	$ 1 + 118.T + 6.24e3T^{2} $
	83	$ 1 - 120.T + 6.88e3T^{2} $
	89	$ 1 - 139. iT - 7.92e3T^{2} $
	97	$ 1 + 44.4iT - 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

−8.582226243088953813473534305454, −7.65537942946866928571659116011, −6.99999824660689704595975434156, −6.53753147037540324798214291442, −5.45503881804264948178290804396, −4.38711221341960374547520683093, −3.90681730474411411006281847568, −2.23065562116152700995132658191, −2.00025660923416413104272992341, −0.38594869503468406554134952352, 0.60403762262717408564160475883, 1.89579752292001798732059246420, 2.76989455420128365280851388704, 3.70518759252377448003170858620, 4.72770077338322700629511353467, 5.82876602553664834895493476089, 6.29668929441101073921353999457, 7.07286942710134815714761364158, 8.147680744226807655657297079360, 8.585879254433313170369595211413

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 $\F_{q}$

\(L(\frac{3}{2})\)	\(\approx\)	\(0.8674480170\)
\(L(\frac12)\)	\(\approx\)	\(0.8674480170\)
\(L(2)\)		not available
\(L(1)\)		not available

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