Properties

Label 2-483-161.18-c1-0-23
Degree $2$
Conductor $483$
Sign $0.557 + 0.830i$
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  + (0.611 + 2.52i)2-s + (−0.327 − 0.945i)3-s + (−4.20 + 2.16i)4-s + (−1.78 + 0.716i)5-s + (2.18 − 1.40i)6-s + (−0.982 − 2.45i)7-s + (−4.64 − 5.36i)8-s + (−0.786 + 0.618i)9-s + (−2.90 − 4.07i)10-s + (0.0659 − 0.271i)11-s + (3.42 + 3.26i)12-s + (−2.23 + 4.90i)13-s + (5.59 − 3.97i)14-s + (1.26 + 1.45i)15-s + (5.18 − 7.28i)16-s + (−0.356 − 7.47i)17-s + ⋯
L(s)  = 1  + (0.432 + 1.78i)2-s + (−0.188 − 0.545i)3-s + (−2.10 + 1.08i)4-s + (−0.800 + 0.320i)5-s + (0.891 − 0.572i)6-s + (−0.371 − 0.928i)7-s + (−1.64 − 1.89i)8-s + (−0.262 + 0.206i)9-s + (−0.917 − 1.28i)10-s + (0.0198 − 0.0819i)11-s + (0.989 + 0.943i)12-s + (−0.621 + 1.36i)13-s + (1.49 − 1.06i)14-s + (0.325 + 0.376i)15-s + (1.29 − 1.82i)16-s + (−0.0863 − 1.81i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.557 + 0.830i$
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: \((2,\ 483,\ (\ :1/2),\ 0.557 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.174448 - 0.0930238i\)
\(L(\frac12)\) \(\approx\) \(0.174448 - 0.0930238i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.982 + 2.45i)T \)
23 \( 1 + (-4.64 + 1.19i)T \)
good2 \( 1 + (-0.611 - 2.52i)T + (-1.77 + 0.916i)T^{2} \)
5 \( 1 + (1.78 - 0.716i)T + (3.61 - 3.45i)T^{2} \)
11 \( 1 + (-0.0659 + 0.271i)T + (-9.77 - 5.04i)T^{2} \)
13 \( 1 + (2.23 - 4.90i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (0.356 + 7.47i)T + (-16.9 + 1.61i)T^{2} \)
19 \( 1 + (-0.396 + 8.31i)T + (-18.9 - 1.80i)T^{2} \)
29 \( 1 + (2.27 - 1.46i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (8.99 - 1.73i)T + (28.7 - 11.5i)T^{2} \)
37 \( 1 + (7.03 - 5.53i)T + (8.72 - 35.9i)T^{2} \)
41 \( 1 + (-0.366 + 2.54i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (4.28 - 4.94i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 + (1.80 + 3.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.42 + 0.136i)T + (52.0 + 10.0i)T^{2} \)
59 \( 1 + (-3.13 - 4.40i)T + (-19.2 + 55.7i)T^{2} \)
61 \( 1 + (3.02 - 8.75i)T + (-47.9 - 37.7i)T^{2} \)
67 \( 1 + (-5.66 + 5.40i)T + (3.18 - 66.9i)T^{2} \)
71 \( 1 + (4.75 + 1.39i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (-1.43 + 0.738i)T + (42.3 - 59.4i)T^{2} \)
79 \( 1 + (2.16 - 0.206i)T + (77.5 - 14.9i)T^{2} \)
83 \( 1 + (-0.193 - 1.34i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (3.47 + 0.669i)T + (82.6 + 33.0i)T^{2} \)
97 \( 1 + (-2.24 + 15.6i)T + (-93.0 - 27.3i)T^{2} \)
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

−11.10688041752405732746233451850, −9.440179399262636756888414538089, −8.828318282773370417463469784748, −7.43175876328722671155004594560, −7.06485000901214750166254704480, −6.77804947923962729877411438727, −5.15897291159966240479103665307, −4.50622300403911473308772300956, −3.23359559016218800856566780201, −0.10489122095830928114595410569, 1.90757320491228290700931923917, 3.40079250642385477526009694485, 3.85709423823792589557773303294, 5.20557188776845739196336006275, 5.81372779350625097976806203596, 7.952385465981176387652754162290, 8.764066139095085090053436560701, 9.750136441379873904501975047587, 10.40472085779821277921883185098, 11.13295491650711192209748319500

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