Properties

Label 2-483-161.18-c1-0-12
Degree $2$
Conductor $483$
Sign $-0.856 + 0.516i$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.632 + 2.60i)2-s + (0.327 + 0.945i)3-s + (−4.62 + 2.38i)4-s + (1.95 − 0.784i)5-s + (−2.25 + 1.45i)6-s + (0.712 + 2.54i)7-s + (−5.63 − 6.49i)8-s + (−0.786 + 0.618i)9-s + (3.28 + 4.61i)10-s + (−1.30 + 5.36i)11-s + (−3.76 − 3.59i)12-s + (0.558 − 1.22i)13-s + (−6.19 + 3.47i)14-s + (1.38 + 1.59i)15-s + (7.35 − 10.3i)16-s + (−0.291 − 6.11i)17-s + ⋯
L(s)  = 1  + (0.447 + 1.84i)2-s + (0.188 + 0.545i)3-s + (−2.31 + 1.19i)4-s + (0.876 − 0.350i)5-s + (−0.921 + 0.592i)6-s + (0.269 + 0.963i)7-s + (−1.99 − 2.29i)8-s + (−0.262 + 0.206i)9-s + (1.03 + 1.45i)10-s + (−0.392 + 1.61i)11-s + (−1.08 − 1.03i)12-s + (0.154 − 0.339i)13-s + (−1.65 + 0.927i)14-s + (0.356 + 0.411i)15-s + (1.83 − 2.58i)16-s + (−0.0706 − 1.48i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.856 + 0.516i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (340, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.856 + 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.470582 - 1.69216i\)
\(L(\frac12)\) \(\approx\) \(0.470582 - 1.69216i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.327 - 0.945i)T \)
7 \( 1 + (-0.712 - 2.54i)T \)
23 \( 1 + (-1.46 - 4.56i)T \)
good2 \( 1 + (-0.632 - 2.60i)T + (-1.77 + 0.916i)T^{2} \)
5 \( 1 + (-1.95 + 0.784i)T + (3.61 - 3.45i)T^{2} \)
11 \( 1 + (1.30 - 5.36i)T + (-9.77 - 5.04i)T^{2} \)
13 \( 1 + (-0.558 + 1.22i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (0.291 + 6.11i)T + (-16.9 + 1.61i)T^{2} \)
19 \( 1 + (-0.188 + 3.95i)T + (-18.9 - 1.80i)T^{2} \)
29 \( 1 + (-2.23 + 1.43i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (-5.25 + 1.01i)T + (28.7 - 11.5i)T^{2} \)
37 \( 1 + (6.68 - 5.25i)T + (8.72 - 35.9i)T^{2} \)
41 \( 1 + (-0.829 + 5.76i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-5.84 + 6.74i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + (-5.81 - 10.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.5 - 1.00i)T + (52.0 + 10.0i)T^{2} \)
59 \( 1 + (-0.116 - 0.164i)T + (-19.2 + 55.7i)T^{2} \)
61 \( 1 + (0.815 - 2.35i)T + (-47.9 - 37.7i)T^{2} \)
67 \( 1 + (2.75 - 2.63i)T + (3.18 - 66.9i)T^{2} \)
71 \( 1 + (-1.44 - 0.425i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (2.49 - 1.28i)T + (42.3 - 59.4i)T^{2} \)
79 \( 1 + (0.202 - 0.0193i)T + (77.5 - 14.9i)T^{2} \)
83 \( 1 + (0.220 + 1.53i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (2.19 + 0.423i)T + (82.6 + 33.0i)T^{2} \)
97 \( 1 + (0.429 - 2.98i)T + (-93.0 - 27.3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.82975215916986583042616862972, −10.07377818317414265489149319993, −9.281184072703773042832007475139, −8.871676060654403794453713664026, −7.66776794530527675689240908670, −6.94035677862996265442828498202, −5.65653081264677363813404933958, −5.16874461280636509148318583502, −4.43754621248219129668201002987, −2.66087601237477931198528780708, 0.983488134324646664411331645732, 2.11412392891204043447751490373, 3.26127898770809697833260126766, 4.16964484174742200358743143726, 5.59413484084449190387548893764, 6.39376261245631995890855436795, 8.178115673729936307206056479485, 8.829412826755250060269319435493, 10.17743854358072430047267234986, 10.51251566548031259483371223970

Graph of the $Z$-function along the critical line