Properties

Label 2-483-161.66-c1-0-28
Degree $2$
Conductor $483$
Sign $-0.237 + 0.971i$
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  + (1.74 − 1.37i)2-s + (0.814 − 0.580i)3-s + (0.691 − 2.84i)4-s + (0.900 + 0.173i)5-s + (0.625 − 2.13i)6-s + (−2.36 − 1.19i)7-s + (−0.859 − 1.88i)8-s + (0.327 − 0.945i)9-s + (1.80 − 0.933i)10-s + (1.96 − 2.49i)11-s + (−1.08 − 2.72i)12-s + (−1.26 + 1.96i)13-s + (−5.76 + 1.15i)14-s + (0.834 − 0.380i)15-s + (1.12 + 0.581i)16-s + (0.328 − 0.312i)17-s + ⋯
L(s)  = 1  + (1.23 − 0.970i)2-s + (0.470 − 0.334i)3-s + (0.345 − 1.42i)4-s + (0.402 + 0.0776i)5-s + (0.255 − 0.869i)6-s + (−0.892 − 0.451i)7-s + (−0.303 − 0.665i)8-s + (0.109 − 0.315i)9-s + (0.572 − 0.295i)10-s + (0.592 − 0.752i)11-s + (−0.314 − 0.785i)12-s + (−0.350 + 0.545i)13-s + (−1.53 + 0.309i)14-s + (0.215 − 0.0983i)15-s + (0.281 + 0.145i)16-s + (0.0795 − 0.0758i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.237 + 0.971i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.237 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89109 - 2.40986i\)
\(L(\frac12)\) \(\approx\) \(1.89109 - 2.40986i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.36 + 1.19i)T \)
23 \( 1 + (-1.31 - 4.61i)T \)
good2 \( 1 + (-1.74 + 1.37i)T + (0.471 - 1.94i)T^{2} \)
5 \( 1 + (-0.900 - 0.173i)T + (4.64 + 1.85i)T^{2} \)
11 \( 1 + (-1.96 + 2.49i)T + (-2.59 - 10.6i)T^{2} \)
13 \( 1 + (1.26 - 1.96i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (-0.328 + 0.312i)T + (0.808 - 16.9i)T^{2} \)
19 \( 1 + (0.682 + 0.651i)T + (0.904 + 18.9i)T^{2} \)
29 \( 1 + (-1.08 - 0.318i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (0.611 - 6.39i)T + (-30.4 - 5.86i)T^{2} \)
37 \( 1 + (-8.69 - 3.00i)T + (29.0 + 22.8i)T^{2} \)
41 \( 1 + (-4.93 - 4.27i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (5.82 + 2.65i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 + (10.0 - 5.81i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.57 - 0.313i)T + (52.7 - 5.03i)T^{2} \)
59 \( 1 + (0.163 + 0.317i)T + (-34.2 + 48.0i)T^{2} \)
61 \( 1 + (1.64 - 2.30i)T + (-19.9 - 57.6i)T^{2} \)
67 \( 1 + (-4.39 + 10.9i)T + (-48.4 - 46.2i)T^{2} \)
71 \( 1 + (1.01 + 7.07i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (-6.33 - 1.53i)T + (64.8 + 33.4i)T^{2} \)
79 \( 1 + (2.20 + 0.104i)T + (78.6 + 7.50i)T^{2} \)
83 \( 1 + (2.33 + 2.69i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (17.1 - 1.64i)T + (87.3 - 16.8i)T^{2} \)
97 \( 1 + (-1.30 + 1.50i)T + (-13.8 - 96.0i)T^{2} \)
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

−11.02650295635655404289464904932, −9.918853716945182710917131525496, −9.316407582858001193431571387799, −7.977882774811449042771519569130, −6.67053583530830271341152023781, −5.99076743150833150296820301637, −4.66946597156515593458619924548, −3.58700152487592791843234456233, −2.86421180823605322427353760901, −1.48839456483869305409066289675, 2.48584213013371295465991190753, 3.67130034806619943247579747927, 4.59393558790771599215540819037, 5.66273144309600192789117612369, 6.40531552757876770173137089467, 7.32551552338871763202364605079, 8.335425447588853104092472139037, 9.575729374452082151185421746563, 9.987842343754822330826641259351, 11.52743137152605159442875259060

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