↑

Properties

Label 1-1323-1323.1046-r0-0-0
Degree $1$
Conductor $1323$
Sign $-0.618 - 0.785i$
Analytic cond. $6.14398$
Root an. cond. $6.14398$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.661 − 0.749i)2-s + (−0.124 − 0.992i)4-s + (0.270 + 0.962i)5-s + (−0.826 − 0.563i)8-s + (0.900 + 0.433i)10-s + (0.661 − 0.749i)11-s + (0.0249 − 0.999i)13-s + (−0.969 + 0.246i)16-s + (−0.222 − 0.974i)17-s − 19-s + (0.921 − 0.388i)20-s + (−0.124 − 0.992i)22-s + (0.797 − 0.603i)23-s + (−0.853 + 0.521i)25-s + (−0.733 − 0.680i)26-s + ⋯
L(s)  = 1  + (0.661 − 0.749i)2-s + (−0.124 − 0.992i)4-s + (0.270 + 0.962i)5-s + (−0.826 − 0.563i)8-s + (0.900 + 0.433i)10-s + (0.661 − 0.749i)11-s + (0.0249 − 0.999i)13-s + (−0.969 + 0.246i)16-s + (−0.222 − 0.974i)17-s − 19-s + (0.921 − 0.388i)20-s + (−0.124 − 0.992i)22-s + (0.797 − 0.603i)23-s + (−0.853 + 0.521i)25-s + (−0.733 − 0.680i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $-0.618 - 0.785i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1046, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ -0.618 - 0.785i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8636367083 - 1.779826388i\)
\(L(\frac12)\) \(\approx\) \(0.8636367083 - 1.779826388i\)
\(L(1)\) \(\approx\) \(1.230560416 - 0.7653295507i\)
\(L(1)\) \(\approx\) \(1.230560416 - 0.7653295507i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.661 - 0.749i)T \)
5 \( 1 + (0.270 + 0.962i)T \)
11 \( 1 + (0.661 - 0.749i)T \)
13 \( 1 + (0.0249 - 0.999i)T \)
17 \( 1 + (-0.222 - 0.974i)T \)
19 \( 1 - T \)
23 \( 1 + (0.797 - 0.603i)T \)
29 \( 1 + (0.797 + 0.603i)T \)
31 \( 1 + (0.939 - 0.342i)T \)
37 \( 1 + (-0.733 + 0.680i)T \)
41 \( 1 + (-0.969 - 0.246i)T \)
43 \( 1 + (0.698 + 0.715i)T \)
47 \( 1 + (-0.318 - 0.947i)T \)
53 \( 1 + (-0.955 + 0.294i)T \)
59 \( 1 + (-0.411 - 0.911i)T \)
61 \( 1 + (0.124 - 0.992i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (0.733 + 0.680i)T \)
73 \( 1 + (-0.365 - 0.930i)T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (-0.0249 - 0.999i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.20919577632976683448119877218, −20.84379259452697600000734145360, −19.69256372372719923939127236586, −19.09002173733445317134590168868, −17.5884807153069831187388788693, −17.34280296517469824812396891302, −16.686977858331404665244559712403, −15.80017896007099722785708963197, −15.138137740834240727444455787763, −14.29260703966910485909980240998, −13.58476459798905440018585544493, −12.78986130939409565066244705017, −12.21376501088450275630856369834, −11.483303283145440992271649880965, −10.20066429237054423177638310061, −9.10043595927348048113576989997, −8.715735093971993359421050825639, −7.7552704181887220960118164154, −6.697363143127347830163830463391, −6.20683979151587934918146541776, −5.10328456829345363487839964454, −4.40044662707992602632962874395, −3.837793262571056490133075415767, −2.37666107550906557209378436658, −1.4308349242433046126449685842, 0.60800359361929930163624499647, 1.86581907454888097597605789700, 2.95480603120855028642965291690, 3.27603071717555369706330924712, 4.507758079433277853371327105253, 5.347199598252481755810586052101, 6.41517212383074876560312845401, 6.72876181542566413758808722747, 8.167266427784740042930153573258, 9.117021881998256504533899016266, 10.03925228356976668811651573785, 10.71148815982906006962955245977, 11.27758662600873149584124912466, 12.126697396877006625046519862873, 13.01285226820708446840256448014, 13.80499233529764763382434860617, 14.32937158140518947516049822819, 15.14825731541048160194644906881, 15.748770955890559648992435810350, 17.01646853956833582812178912794, 17.80967292360269461977956619759, 18.70591445131385501804324858076, 19.09640797779612000360250014894, 19.99261133078277800814374888098, 20.73226392779076554390338099105

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