Properties

Label 2-483-161.61-c1-0-31
Degree $2$
Conductor $483$
Sign $-0.586 - 0.809i$
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  + (−0.451 − 0.355i)2-s + (−0.814 − 0.580i)3-s + (−0.393 − 1.62i)4-s + (0.598 − 0.115i)5-s + (0.161 + 0.551i)6-s + (−1.36 + 2.26i)7-s + (−0.875 + 1.91i)8-s + (0.327 + 0.945i)9-s + (−0.311 − 0.160i)10-s + (−1.70 − 2.16i)11-s + (−0.620 + 1.55i)12-s + (−2.52 − 3.93i)13-s + (1.42 − 0.536i)14-s + (−0.554 − 0.253i)15-s + (−1.89 + 0.975i)16-s + (0.524 + 0.499i)17-s + ⋯
L(s)  = 1  + (−0.319 − 0.251i)2-s + (−0.470 − 0.334i)3-s + (−0.196 − 0.811i)4-s + (0.267 − 0.0516i)5-s + (0.0660 + 0.224i)6-s + (−0.517 + 0.855i)7-s + (−0.309 + 0.677i)8-s + (0.109 + 0.315i)9-s + (−0.0984 − 0.0507i)10-s + (−0.513 − 0.653i)11-s + (−0.179 + 0.447i)12-s + (−0.700 − 1.09i)13-s + (0.379 − 0.143i)14-s + (−0.143 − 0.0654i)15-s + (−0.473 + 0.243i)16-s + (0.127 + 0.121i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.586 - 0.809i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.586 - 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000885101 + 0.00173458i\)
\(L(\frac12)\) \(\approx\) \(0.000885101 + 0.00173458i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.814 + 0.580i)T \)
7 \( 1 + (1.36 - 2.26i)T \)
23 \( 1 + (2.08 - 4.31i)T \)
good2 \( 1 + (0.451 + 0.355i)T + (0.471 + 1.94i)T^{2} \)
5 \( 1 + (-0.598 + 0.115i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (1.70 + 2.16i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (2.52 + 3.93i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (-0.524 - 0.499i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (3.03 - 2.89i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (-3.39 + 0.996i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-0.714 - 7.48i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (2.76 - 0.956i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (-2.08 + 1.80i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (4.55 - 2.08i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-2.68 - 1.54i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.5 + 0.548i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (-4.15 + 8.05i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (4.87 + 6.83i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (-0.189 - 0.473i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (-0.444 + 3.08i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (8.90 - 2.16i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (9.48 - 0.451i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (-5.91 + 6.82i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-7.15 - 0.683i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (3.39 + 3.92i)T + (-13.8 + 96.0i)T^{2} \)
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

−10.27086988014588278694337373678, −9.777580410489791012553145303104, −8.678107514636493777652655793302, −7.84578269345370627339218173217, −6.35996225028297483150022923905, −5.70994143567864729950942803155, −5.04173301928195161741809038852, −3.08612321458774405357127991593, −1.78146408257721308230476698332, −0.00131369298318033497757899649, 2.51742430206025042884460984709, 4.05973594776658354140929085434, 4.64743521667078995865472819860, 6.29101327341799823403786637879, 7.00871446597876053201142994002, 7.82260942825499255942294227329, 9.015710429808090076216462614084, 9.804584589447258555080037774385, 10.42396424173977271687362111099, 11.61876953700305484399429420086

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