↑

Properties

Label 1-819-819.25-r0-0-0
Degree $1$
Conductor $819$
Sign $0.281 - 0.959i$
Analytic cond. $3.80342$
Root an. cond. $3.80342$
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  − 2-s + 4-s + (0.5 + 0.866i)5-s − 8-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + 16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s − 31-s − 32-s + ⋯
L(s)  = 1  − 2-s + 4-s + (0.5 + 0.866i)5-s − 8-s + (−0.5 − 0.866i)10-s + (0.5 − 0.866i)11-s + 16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s − 31-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.281 - 0.959i$
Analytic conductor: \(3.80342\)
Root analytic conductor: \(3.80342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 819,\ (0:\ ),\ 0.281 - 0.959i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6281997398 - 0.4702196802i\)
\(L(\frac12)\) \(\approx\) \(0.6281997398 - 0.4702196802i\)
\(L(1)\) \(\approx\) \(0.7159253150 - 0.07585233497i\)
\(L(1)\) \(\approx\) \(0.7159253150 - 0.07585233497i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 - T \)
5 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 - T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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

−22.092464482960537730941920584079, −21.43367653966262325588400406139, −20.39467699471079021629340152684, −20.08977003326669458426250660689, −19.2427298697787352959246184438, −18.04076329977665087626099471668, −17.72919480727375386108043561145, −16.708096238410141601286021143058, −16.334159512646485591668285070386, −15.20543727340884362545912745749, −14.501604597466076002031030308119, −13.220683816012317131731531427013, −12.44851410319749442389479971000, −11.70569317744521048816793599567, −10.67336050877633848112692393747, −9.68303343182404904270164962625, −9.32117281722887010038643191258, −8.28483121327801389182141829813, −7.5638504102486263952322365718, −6.44062174675993195836604363802, −5.69319476799713717561541178329, −4.5207213121252259434742243971, −3.31325701802536323761032056848, −1.801520598921224731913256488414, −1.44451678302319553379477590503, 0.48604398234744635069091065528, 1.9608020995484224417425325038, 2.74468418968093552066838160741, 3.73894513086191947212603356150, 5.39321809800213718117359060612, 6.30988227234497744324264412527, 6.97538427441170135026985412430, 7.83957216060468282913282311968, 8.99450622703270546959563610529, 9.48968487007505549174869861094, 10.52688336766182236577132182167, 11.19422452868610749067253251623, 11.80901842197139922092729894042, 13.147387800490764654401424938811, 14.08166293795266308189260817431, 14.81394874258228501059519564912, 15.80162237994380473007973947536, 16.48026434687596838053895720432, 17.426634752673784494341263501133, 18.07939301460038370533346552108, 18.71323474309803808740558353250, 19.47848860314781213558759493926, 20.28441130481371483903178580326, 21.160176741165588814595386109438, 22.014738461368755165125803972359

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