LMFDB - L-function 2-1800-15.14-c2-0-8

Dirichlet series

L(s) = 1

− 3.32i·7^-s + 16.4i·11^-s − 5.64i·13^-s + 13.1·17^-s − 7.97·19^-s + 9.89·23^-s − 6.61i·29^-s − 25.3·31^-s + 37.9i·37^-s − 16.4i·41^-s + 20.6i·43^-s − 13.1·47^-s + 37.9·49^-s − 16.4·53^-s + 6.61i·59^-s + ⋯

L(s) = 1

− 0.474i·7^-s + 1.49i·11^-s − 0.434i·13^-s + 0.775·17^-s − 0.419·19^-s + 0.430·23^-s − 0.228i·29^-s − 0.816·31^-s + 1.02i·37^-s − 0.400i·41^-s + 0.480i·43^-s − 0.280·47^-s + 0.774·49^-s − 0.310·53^-s + 0.112i·59^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(3-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$1800$ = $2^{3} \cdot 3^{2} \cdot 5^{2}$
Sign:	$0.151 - 0.988i$
Analytic conductor:	$49.0464$
Root analytic conductor:	$7.00331$
Motivic weight:	$2$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{1800} (449, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 1800,\ (\ :1),\ 0.151 - 0.988i)$

Particular Values

$L(\frac{3}{2})$	$\approx$	$1.526258063$
$L(\frac12)$	$\approx$	$1.526258063$
$L(2)$		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 $
	3	$ 1 $
	5	$ 1 $
good	7	$ 1 + 3.32iT - 49T^{2} $
	11	$ 1 - 16.4iT - 121T^{2} $
	13	$ 1 + 5.64iT - 169T^{2} $
	17	$ 1 - 13.1T + 289T^{2} $
	19	$ 1 + 7.97T + 361T^{2} $
	23	$ 1 - 9.89T + 529T^{2} $
	29	$ 1 + 6.61iT - 841T^{2} $
	31	$ 1 + 25.3T + 961T^{2} $
	37	$ 1 - 37.9iT - 1.36e3T^{2} $
	41	$ 1 + 16.4iT - 1.68e3T^{2} $
	43	$ 1 - 20.6iT - 1.84e3T^{2} $
	47	$ 1 + 13.1T + 2.20e3T^{2} $
	53	$ 1 + 16.4T + 2.80e3T^{2} $
	59	$ 1 - 6.61iT - 3.48e3T^{2} $
	61	$ 1 + 104.T + 3.72e3T^{2} $
	67	$ 1 - 64.0iT - 4.48e3T^{2} $
	71	$ 1 - 58.0iT - 5.04e3T^{2} $
	73	$ 1 - 128. iT - 5.32e3T^{2} $
	79	$ 1 - 90.5T + 6.24e3T^{2} $
	83	$ 1 - 80.9T + 6.88e3T^{2} $
	89	$ 1 + 74.4iT - 7.92e3T^{2} $
	97	$ 1 + 69.4iT - 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

−9.400208879025559767107492215560, −8.429440227844883071795688190669, −7.54828885859760253097669073343, −7.08236960244962588872490865137, −6.09258117489023790144882701151, −5.10150202056436738304923543412, −4.37030794902455735129306695380, −3.40116137029780002120031695683, −2.25583872325006563099288431821, −1.12221809464485687203855443612, 0.43523168052307160533947704003, 1.77634946063383515519413588050, 3.00074490864400932297010386885, 3.73020414618958513463942196331, 4.92482837553284975688479829193, 5.76440307244540043121324476390, 6.35473491321161943430462700079, 7.41813631972037548526810725797, 8.189474229864594771047892095809, 8.982120155582279998323403734009

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

\(L(\frac{3}{2})\)	\(\approx\)	\(1.526258063\)
\(L(\frac12)\)	\(\approx\)	\(1.526258063\)
\(L(2)\)		not available
\(L(1)\)		not available

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