↑

Properties

Label 1-192-192.155-r0-0-0
Degree $1$
Conductor $192$
Sign $0.881 + 0.471i$
Analytic cond. $0.891644$
Root an. cond. $0.891644$
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  + (0.923 + 0.382i)5-s + (0.707 + 0.707i)7-s + (0.382 − 0.923i)11-s + (−0.923 + 0.382i)13-s − i·17-s + (−0.923 + 0.382i)19-s + (0.707 − 0.707i)23-s + (0.707 + 0.707i)25-s + (−0.382 − 0.923i)29-s + 31-s + (0.382 + 0.923i)35-s + (0.923 + 0.382i)37-s + (−0.707 + 0.707i)41-s + (0.382 − 0.923i)43-s − i·47-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)5-s + (0.707 + 0.707i)7-s + (0.382 − 0.923i)11-s + (−0.923 + 0.382i)13-s − i·17-s + (−0.923 + 0.382i)19-s + (0.707 − 0.707i)23-s + (0.707 + 0.707i)25-s + (−0.382 − 0.923i)29-s + 31-s + (0.382 + 0.923i)35-s + (0.923 + 0.382i)37-s + (−0.707 + 0.707i)41-s + (0.382 − 0.923i)43-s − i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.881 + 0.471i$
Analytic conductor: \(0.891644\)
Root analytic conductor: \(0.891644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 192,\ (0:\ ),\ 0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.354879704 + 0.3393796986i\)
\(L(\frac12)\) \(\approx\) \(1.354879704 + 0.3393796986i\)
\(L(1)\) \(\approx\) \(1.240731043 + 0.1672287028i\)
\(L(1)\) \(\approx\) \(1.240731043 + 0.1672287028i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.923 + 0.382i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (-0.382 - 0.923i)T \)
31 \( 1 + T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (0.382 - 0.923i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (-0.923 - 0.382i)T \)
61 \( 1 + (-0.382 - 0.923i)T \)
67 \( 1 + (0.382 + 0.923i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (0.923 - 0.382i)T \)
89 \( 1 + (-0.707 - 0.707i)T \)
97 \( 1 - T \)
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

−27.154990936601694676887972890106, −25.89543829806675868996425943224, −25.08218705527335237678632311621, −24.28682684443930082252668674226, −23.205335961175777265434247918222, −22.19754854874473150656829306922, −21.14362496992834286425022663355, −20.41628175679396621058852900575, −19.50917173821257200917770506208, −17.944566121343356694754409673809, −17.42030092297971686719003485128, −16.6197053822447124028665450891, −15.100057906540946783318400436364, −14.27408023861044748347669299928, −13.27606324971317362204747542875, −12.30606234766875136246902877204, −11.02166971309174151440520719687, −9.93635944739128822692676013945, −9.111823810623583868444006822686, −7.673382926527782029982932579615, −6.72119619170361718472003610807, −5.184623672214320195104273873247, −4.4721789845187827267664585411, −2.58664265543758220221317008900, −1.32051166935148870056533084229, 1.720704888050344803203243826650, 2.775612350867212619222820449, 4.47321426346470522029609594137, 5.75509633703940357206827133098, 6.54758630029331667435074096438, 8.13244949228574138884443443938, 9.05759626401562241446905163966, 10.23051245627519766728830052759, 11.23002898431840524714552734269, 12.33226421945730943603408378609, 13.51207964459498084604796692962, 14.54867629786509764743687445630, 15.133074352636204591378537038381, 16.86213480582607762229649255712, 17.29521519804917957386495579167, 18.63484341506707788615086389438, 19.15414270035112177769343400799, 20.69891236485616147812363862542, 21.67553868939491237007915758326, 21.947439322889388464165983428679, 23.40544036990783667980690255242, 24.60527514229585634504515891290, 24.99127157755568101941350595935, 26.28251445815524681407190004739, 27.011016786141351530399335094356

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