↑

Properties

Label 1-384-384.173-r1-0-0
Degree $1$
Conductor $384$
Sign $-0.146 + 0.989i$
Analytic cond. $41.2665$
Root an. cond. $41.2665$
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  + (−0.195 − 0.980i)5-s + (0.382 + 0.923i)7-s + (0.831 + 0.555i)11-s + (−0.195 + 0.980i)13-s + (−0.707 − 0.707i)17-s + (0.980 + 0.195i)19-s + (−0.923 − 0.382i)23-s + (−0.923 + 0.382i)25-s + (−0.831 + 0.555i)29-s − i·31-s + (0.831 − 0.555i)35-s + (−0.980 + 0.195i)37-s + (0.923 + 0.382i)41-s + (−0.555 + 0.831i)43-s + (0.707 + 0.707i)47-s + ⋯
L(s)  = 1  + (−0.195 − 0.980i)5-s + (0.382 + 0.923i)7-s + (0.831 + 0.555i)11-s + (−0.195 + 0.980i)13-s + (−0.707 − 0.707i)17-s + (0.980 + 0.195i)19-s + (−0.923 − 0.382i)23-s + (−0.923 + 0.382i)25-s + (−0.831 + 0.555i)29-s − i·31-s + (0.831 − 0.555i)35-s + (−0.980 + 0.195i)37-s + (0.923 + 0.382i)41-s + (−0.555 + 0.831i)43-s + (0.707 + 0.707i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $-0.146 + 0.989i$
Analytic conductor: \(41.2665\)
Root analytic conductor: \(41.2665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (1:\ ),\ -0.146 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8494667993 + 0.9847680934i\)
\(L(\frac12)\) \(\approx\) \(0.8494667993 + 0.9847680934i\)
\(L(1)\) \(\approx\) \(0.9935894457 + 0.1277062515i\)
\(L(1)\) \(\approx\) \(0.9935894457 + 0.1277062515i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.195 - 0.980i)T \)
7 \( 1 + (0.382 + 0.923i)T \)
11 \( 1 + (0.831 + 0.555i)T \)
13 \( 1 + (-0.195 + 0.980i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (0.980 + 0.195i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
29 \( 1 + (-0.831 + 0.555i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.980 + 0.195i)T \)
41 \( 1 + (0.923 + 0.382i)T \)
43 \( 1 + (-0.555 + 0.831i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (-0.831 - 0.555i)T \)
59 \( 1 + (0.195 + 0.980i)T \)
61 \( 1 + (0.555 + 0.831i)T \)
67 \( 1 + (0.555 + 0.831i)T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (0.980 + 0.195i)T \)
89 \( 1 + (-0.923 + 0.382i)T \)
97 \( 1 - iT \)
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

−24.06353200692634411873955972091, −23.10928196977402627132577776958, −22.30487604781299840386702559977, −21.707006479205736737306401528618, −20.32968328128344867013127439958, −19.780883406024565431888298094185, −18.85474751965396596211100829640, −17.72174438040241194362043327994, −17.29123501128470727094337510228, −15.9857888557011172883050417808, −15.14333056257950627482484428208, −14.14361900907669470314743960291, −13.64757177289614470501269591993, −12.27480666160027604884774685225, −11.21382770630856676742865131734, −10.63387139168082463822528237662, −9.660394325069298705411806239568, −8.29681797680845842586014473811, −7.42588876884751244773602127176, −6.57539141692275592594398747482, −5.45548719722824017117046910943, −3.97764549977294437545856728942, −3.326478719143932456917153894536, −1.82888566794414624248853662380, −0.365186087812493308909852664904, 1.329158168843393948381027518069, 2.31923578313799747389532876411, 4.00466062801701115383749803606, 4.81692483366139454981603719998, 5.819962604455393695937763414180, 7.047714643539734073236357188978, 8.164535493246209061198592494766, 9.1792358046389593517269250908, 9.58615424792000973744203436484, 11.41728776807342241890257087855, 11.87478387833915512405967082656, 12.724242397787701563320510710651, 13.89859546356668796707001590224, 14.75734626567133321553731687347, 15.82611951519129470158060389076, 16.486673797037204148932507540455, 17.51453385533374267564260844111, 18.34442848546848511227199088046, 19.36763067306553096348850950800, 20.278761580017778881068276894515, 20.91004640842312466967429816596, 22.02269458381120262017742471899, 22.611618341472706202098172704516, 24.05300473905816918831885141401, 24.412735988081829606858418111156

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