LMFDB - L-function 2-45e2-5.4-c1-0-42

Dirichlet series

L(s) = 1

− 2.63i·2^-s − 4.93·4^-s + 1.79i·7^-s + 7.73i·8^-s + 1.80·11^-s + 1.97i·13^-s + 4.73·14^-s + 10.4·16^-s − 4.80i·17^-s − 2.96·19^-s − 4.76i·22^-s − 1.73i·23^-s + 5.19·26^-s − 8.87i·28^-s + 7.36·29^-s + ⋯

L(s) = 1

− 1.86i·2^-s − 2.46·4^-s + 0.679i·7^-s + 2.73i·8^-s + 0.545·11^-s + 0.546i·13^-s + 1.26·14^-s + 2.62·16^-s − 1.16i·17^-s − 0.680·19^-s − 1.01i·22^-s − 0.361i·23^-s + 1.01·26^-s − 1.67i·28^-s + 1.36·29^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	$2$
Conductor:	$2025$ = $3^{4} \cdot 5^{2}$
Sign:	$-0.894 - 0.447i$
Analytic conductor:	$16.1697$
Root analytic conductor:	$4.02115$
Motivic weight:	$1$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2025} (649, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(2,\ 2025,\ (\ :1/2),\ -0.894 - 0.447i)$

Particular Values

$L(1)$	$\approx$	$1.044625796$
$L(\frac12)$	$\approx$	$1.044625796$
$L(\frac{3}{2})$		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 + 2.63iT - 2T^{2} $
	7	$ 1 - 1.79iT - 7T^{2} $
	11	$ 1 - 1.80T + 11T^{2} $
	13	$ 1 - 1.97iT - 13T^{2} $
	17	$ 1 + 4.80iT - 17T^{2} $
	19	$ 1 + 2.96T + 19T^{2} $
	23	$ 1 + 1.73iT - 23T^{2} $
	29	$ 1 - 7.36T + 29T^{2} $
	31	$ 1 + 2.62T + 31T^{2} $
	37	$ 1 + 11.6iT - 37T^{2} $
	41	$ 1 + 2.46T + 41T^{2} $
	43	$ 1 + 7.27iT - 43T^{2} $
	47	$ 1 + 6.29iT - 47T^{2} $
	53	$ 1 - 1.72iT - 53T^{2} $
	59	$ 1 + 11.0T + 59T^{2} $
	61	$ 1 + 12.6T + 61T^{2} $
	67	$ 1 + 9.10iT - 67T^{2} $
	71	$ 1 - 1.27T + 71T^{2} $
	73	$ 1 + 3.58iT - 73T^{2} $
	79	$ 1 + 2.11T + 79T^{2} $
	83	$ 1 - 1.09iT - 83T^{2} $
	89	$ 1 - 13.2T + 89T^{2} $
	97	$ 1 - 3.83iT - 97T^{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

−9.103427155744517579577795087346, −8.442361147908037910746983459542, −7.23953079458372765115103253581, −6.11509494671069215999066192522, −5.06108919461153251157084590245, −4.37575561070682940959708402838, −3.47278880662361088884504973645, −2.53200072005461911125476286815, −1.81171299347363335788793195324, −0.42769262030207582986946232876, 1.20091999660649536534909648677, 3.29680302217456312313001041021, 4.29537733061498390332381056362, 4.83277693843911881567078170177, 6.02709166143540766807432708749, 6.36537439409205779240734560638, 7.20020733686352957527244154619, 7.988698792180639581169843830214, 8.445662986407929651388134160517, 9.274551173281803838282029428588

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.044625796\)
\(L(\frac12)\)	\(\approx\)	\(1.044625796\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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