↑

Properties

Label 1-648-648.211-r1-0-0
Degree $1$
Conductor $648$
Sign $-0.565 + 0.824i$
Analytic cond. $69.6372$
Root an. cond. $69.6372$
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  + (0.686 + 0.727i)5-s + (0.993 − 0.116i)7-s + (−0.286 − 0.957i)11-s + (0.0581 + 0.998i)13-s + (−0.939 − 0.342i)17-s + (−0.939 + 0.342i)19-s + (0.993 + 0.116i)23-s + (−0.0581 + 0.998i)25-s + (−0.893 + 0.448i)29-s + (−0.597 + 0.802i)31-s + (0.766 + 0.642i)35-s + (−0.766 + 0.642i)37-s + (−0.835 + 0.549i)41-s + (0.973 + 0.230i)43-s + (−0.597 − 0.802i)47-s + ⋯
L(s)  = 1  + (0.686 + 0.727i)5-s + (0.993 − 0.116i)7-s + (−0.286 − 0.957i)11-s + (0.0581 + 0.998i)13-s + (−0.939 − 0.342i)17-s + (−0.939 + 0.342i)19-s + (0.993 + 0.116i)23-s + (−0.0581 + 0.998i)25-s + (−0.893 + 0.448i)29-s + (−0.597 + 0.802i)31-s + (0.766 + 0.642i)35-s + (−0.766 + 0.642i)37-s + (−0.835 + 0.549i)41-s + (0.973 + 0.230i)43-s + (−0.597 − 0.802i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.565 + 0.824i$
Analytic conductor: \(69.6372\)
Root analytic conductor: \(69.6372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 648,\ (1:\ ),\ -0.565 + 0.824i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7642094873 + 1.450815923i\)
\(L(\frac12)\) \(\approx\) \(0.7642094873 + 1.450815923i\)
\(L(1)\) \(\approx\) \(1.116464549 + 0.2927379409i\)
\(L(1)\) \(\approx\) \(1.116464549 + 0.2927379409i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.686 + 0.727i)T \)
7 \( 1 + (0.993 - 0.116i)T \)
11 \( 1 + (-0.286 - 0.957i)T \)
13 \( 1 + (0.0581 + 0.998i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (0.993 + 0.116i)T \)
29 \( 1 + (-0.893 + 0.448i)T \)
31 \( 1 + (-0.597 + 0.802i)T \)
37 \( 1 + (-0.766 + 0.642i)T \)
41 \( 1 + (-0.835 + 0.549i)T \)
43 \( 1 + (0.973 + 0.230i)T \)
47 \( 1 + (-0.597 - 0.802i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.286 + 0.957i)T \)
61 \( 1 + (-0.396 + 0.918i)T \)
67 \( 1 + (0.893 + 0.448i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (0.835 + 0.549i)T \)
83 \( 1 + (-0.835 - 0.549i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (-0.686 + 0.727i)T \)
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

−22.3231490386572471508944965461, −21.44068249804023065847256772635, −20.617458972646359266521978768143, −20.307973935765605296047567383706, −19.090302289289628158880761165539, −17.95986491411456466201782437093, −17.47743752632803477624004876033, −16.9078396960310126617862186324, −15.493296208292877389030475936086, −15.05767895622654739088863145417, −14.01163514202181099468402381438, −12.86992460256406925080287328425, −12.70088007625561917717311697837, −11.26555966562817824409381500716, −10.587959407802432152540754965789, −9.524961300887804264118000142141, −8.6897970052684151397461201609, −7.90684494679796005358296770880, −6.82178388399145948621228047651, −5.59753667071196092338777076904, −4.95702305943147800726701152967, −4.069603368364566038809460565894, −2.36884144688650534243979228108, −1.743652713410912484447479232767, −0.353574618834297719106370415549, 1.41384981218445051241245356702, 2.27583882757446698121276186566, 3.41789773705519393356873386964, 4.61716624969742878338717633363, 5.55683673001570232063162871362, 6.572350996075039130483442012775, 7.31384677945581338870930065380, 8.57937197008908822407537071577, 9.16307342114621234553631976747, 10.55521235452859559388849056670, 10.969733544004600344462091870221, 11.78862805592397522910384206252, 13.18429064803337248885085566341, 13.800966291414624934065790718050, 14.57671871534358776174203877222, 15.26962456122654759387167231520, 16.54515856907986328826265498126, 17.15958381732415879317501013308, 18.16111301841913100639941304266, 18.65142523004166869904785425490, 19.55380198962548448349005075920, 20.781286978766264853787908431469, 21.36546865733032969429608052655, 21.910703018658129940954589841803, 22.951096318143370187428511439641

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