Properties

Label 1-297-297.212-r1-0-0
Degree $1$
Conductor $297$
Sign $0.847 - 0.530i$
Analytic cond. $31.9170$
Root an. cond. $31.9170$
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  + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.0348 + 0.999i)5-s + (−0.719 + 0.694i)7-s + (0.104 + 0.994i)8-s + (−0.5 − 0.866i)10-s + (−0.374 − 0.927i)13-s + (0.241 − 0.970i)14-s + (−0.615 − 0.788i)16-s + (−0.669 + 0.743i)17-s + (−0.104 − 0.994i)19-s + (0.882 + 0.469i)20-s + (−0.173 − 0.984i)23-s + (−0.997 − 0.0697i)25-s + (0.809 + 0.587i)26-s + ⋯
L(s)  = 1  + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.0348 + 0.999i)5-s + (−0.719 + 0.694i)7-s + (0.104 + 0.994i)8-s + (−0.5 − 0.866i)10-s + (−0.374 − 0.927i)13-s + (0.241 − 0.970i)14-s + (−0.615 − 0.788i)16-s + (−0.669 + 0.743i)17-s + (−0.104 − 0.994i)19-s + (0.882 + 0.469i)20-s + (−0.173 − 0.984i)23-s + (−0.997 − 0.0697i)25-s + (0.809 + 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.847 - 0.530i$
Analytic conductor: \(31.9170\)
Root analytic conductor: \(31.9170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ 0.847 - 0.530i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5291629338 - 0.1518790282i\)
\(L(\frac12)\) \(\approx\) \(0.5291629338 - 0.1518790282i\)
\(L(1)\) \(\approx\) \(0.5585966970 + 0.1792839368i\)
\(L(1)\) \(\approx\) \(0.5585966970 + 0.1792839368i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.848 + 0.529i)T \)
5 \( 1 + (-0.0348 + 0.999i)T \)
7 \( 1 + (-0.719 + 0.694i)T \)
13 \( 1 + (-0.374 - 0.927i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (0.241 + 0.970i)T \)
31 \( 1 + (0.990 + 0.139i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
41 \( 1 + (0.241 - 0.970i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.438 - 0.898i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.559 - 0.829i)T \)
61 \( 1 + (0.990 - 0.139i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.848 - 0.529i)T \)
83 \( 1 + (0.374 - 0.927i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.0348 + 0.999i)T \)
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

−25.30466674611647557098835736886, −24.62879492162768287586466525183, −23.50767926791544852651826052173, −22.465304241062551025451192001972, −21.28129286098762862119368279121, −20.675618041323692731621980907817, −19.65408252894414040864038615813, −19.2587835077989906516414969713, −17.936339270968843152368031116903, −17.02295458764274406208055471853, −16.39384408835827241700256029018, −15.65613838548275219418450718496, −13.86805246568194231789252728576, −13.065925239253037269818189309144, −12.09192264348285285571760890073, −11.311669047880379360163039516265, −9.88268206327968125482681204593, −9.48977618758922349547382531112, −8.33762248015045316902648464104, −7.3731116356369679626463175284, −6.28033774911158425868643855324, −4.571669188248980435737694938976, −3.64418627785404642691601311376, −2.14554895080026974873380872453, −0.90442846328270540877020028111, 0.27416908476070425357687811952, 2.2035578183857282389801979840, 3.11788081687674376299760451782, 5.021204277307472179343047500168, 6.30703683004392930924978740217, 6.773666534893378928656927455809, 8.04924470276951539605094054255, 8.96430810662532511730887492374, 10.11172087080209192766344224224, 10.69592364308179381557122026414, 11.87127865575593986197899475987, 13.13056832801745210314617091673, 14.41211970673721802161270975948, 15.300033825796009426937946042825, 15.72840223182690921866080565820, 17.05982656295604891863124045260, 17.89686480320907160228664523357, 18.65189382025353702211066820972, 19.4585609748883527767056689076, 20.1785424462479798155542677847, 21.75872730208089517233192370208, 22.43294671887878675208187646112, 23.378502367869643951597247751333, 24.47199587379247456170416309041, 25.27276490793583586340432232839

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