LMFDB - L-function 2-45e2-5.4-c1-0-57

Dirichlet series

L(s) = 1

− 1.30i·2^-s + 0.302·4^-s + 0.697i·7^-s − 3i·8^-s + 1.69·11^-s − 3.30i·13^-s + 0.908·14^-s − 3.30·16^-s − 1.30i·17^-s − 7.21·19^-s − 2.21i·22^-s − 3.90i·23^-s − 4.30·26^-s + 0.211i·28^-s + 8.60·29^-s + ⋯

L(s) = 1

− 0.921i·2^-s + 0.151·4^-s + 0.263i·7^-s − 1.06i·8^-s + 0.511·11^-s − 0.916i·13^-s + 0.242·14^-s − 0.825·16^-s − 0.315i·17^-s − 1.65·19^-s − 0.471i·22^-s − 0.814i·23^-s − 0.843·26^-s + 0.0398i·28^-s + 1.59·29^-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

$L(1)$	$\approx$	$1.654899217$
$L(\frac12)$	$\approx$	$1.654899217$
$L(\frac{3}{2})$		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 + 1.30iT - 2T^{2} $
	7	$ 1 - 0.697iT - 7T^{2} $
	11	$ 1 - 1.69T + 11T^{2} $
	13	$ 1 + 3.30iT - 13T^{2} $
	17	$ 1 + 1.30iT - 17T^{2} $
	19	$ 1 + 7.21T + 19T^{2} $
	23	$ 1 + 3.90iT - 23T^{2} $
	29	$ 1 - 8.60T + 29T^{2} $
	31	$ 1 + 6.21T + 31T^{2} $
	37	$ 1 - 8.90iT - 37T^{2} $
	41	$ 1 + 1.69T + 41T^{2} $
	43	$ 1 + 9.30iT - 43T^{2} $
	47	$ 1 + 8.21iT - 47T^{2} $
	53	$ 1 + 12.5iT - 53T^{2} $
	59	$ 1 - 7.30T + 59T^{2} $
	61	$ 1 - 3.30T + 61T^{2} $
	67	$ 1 - 1.60iT - 67T^{2} $
	71	$ 1 + 11.6T + 71T^{2} $
	73	$ 1 + 2.39iT - 73T^{2} $
	79	$ 1 + 4.60T + 79T^{2} $
	83	$ 1 - 9iT - 83T^{2} $
	89	$ 1 + 12T + 89T^{2} $
	97	$ 1 - 2iT - 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

−8.725024134728237474747741252952, −8.381533110979146771187497273355, −7.03433249089554865262312949563, −6.59249079773301988841752214896, −5.59684894950298703453432645473, −4.51204048317566025274591929269, −3.63742358483366699254908321049, −2.71636473690532651063242194808, −1.90046766099573227989738467347, −0.56231359545048701020268904115, 1.56463826257500918017741903753, 2.59404033546196957090477960331, 3.97801879871211660199645154615, 4.65457795651449571383251106186, 5.84150669924327741065101993290, 6.34632658997704989683752427787, 7.09711898663331385351979832167, 7.72099257249090425975849745514, 8.689276652449482413480469702646, 9.106403953868416935261799979618

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.654899217\)
\(L(\frac12)\)	\(\approx\)	\(1.654899217\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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