LMFDB - L-function 2-45e2-5.4-c1-0-52

Dirichlet series

L(s) = 1

− 1.63i·2^-s − 0.669·4^-s − 0.505i·7^-s − 2.17i·8^-s + 3.10·11^-s − 6.23i·13^-s − 0.825·14^-s − 4.89·16^-s + 6.10i·17^-s + 5.57·19^-s − 5.06i·22^-s − 3.82i·23^-s − 10.1·26^-s + 0.338i·28^-s − 2.45·29^-s + ⋯

L(s) = 1

− 1.15i·2^-s − 0.334·4^-s − 0.191i·7^-s − 0.768i·8^-s + 0.934·11^-s − 1.73i·13^-s − 0.220·14^-s − 1.22·16^-s + 1.47i·17^-s + 1.27·19^-s − 1.07i·22^-s − 0.797i·23^-s − 1.99·26^-s + 0.0638i·28^-s − 0.456·29^-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

$L(1)$	$\approx$	$1.920688350$
$L(\frac12)$	$\approx$	$1.920688350$
$L(\frac{3}{2})$		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 + 1.63iT - 2T^{2} $
	7	$ 1 + 0.505iT - 7T^{2} $
	11	$ 1 - 3.10T + 11T^{2} $
	13	$ 1 + 6.23iT - 13T^{2} $
	17	$ 1 - 6.10iT - 17T^{2} $
	19	$ 1 - 5.57T + 19T^{2} $
	23	$ 1 + 3.82iT - 23T^{2} $
	29	$ 1 + 2.45T + 29T^{2} $
	31	$ 1 - 4.22T + 31T^{2} $
	37	$ 1 + 6.72iT - 37T^{2} $
	41	$ 1 + 5.44T + 41T^{2} $
	43	$ 1 - 1.32iT - 43T^{2} $
	47	$ 1 - 3.70iT - 47T^{2} $
	53	$ 1 - 2.54iT - 53T^{2} $
	59	$ 1 + 2.88T + 59T^{2} $
	61	$ 1 + 2.84T + 61T^{2} $
	67	$ 1 + 2.40iT - 67T^{2} $
	71	$ 1 - 5.54T + 71T^{2} $
	73	$ 1 + 11.7iT - 73T^{2} $
	79	$ 1 + 3.40T + 79T^{2} $
	83	$ 1 + 13.9iT - 83T^{2} $
	89	$ 1 + 3.38T + 89T^{2} $
	97	$ 1 + 11.0iT - 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

−9.034348626760276112955625819774, −8.122397974260851623690740922285, −7.33999217938790386824635381114, −6.36375576129382308338160679636, −5.62189493455816455901493116862, −4.40330579365661897029433689559, −3.56803492592950249179983138441, −2.91769773678406536251495639913, −1.69143183919361226886144494268, −0.72575795276903801257747343921, 1.46261698546784323059281181563, 2.68505374313997770400563664893, 3.92880529227357721177855879555, 4.92140266173881322183223189813, 5.55347396633613289782871439938, 6.68630380065311129013695989227, 6.88778168837358551127769524564, 7.70637712566014949040932551075, 8.658029531172382700184778999345, 9.313845758537882002822121641334

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 $\F_{q}$

\(L(1)\)	\(\approx\)	\(1.920688350\)
\(L(\frac12)\)	\(\approx\)	\(1.920688350\)
\(L(\frac{3}{2})\)		not available
\(L(1)\)		not available

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