Properties

Label 1-320-320.189-r0-0-0
Degree $1$
Conductor $320$
Sign $0.471 + 0.881i$
Analytic cond. $1.48607$
Root an. cond. $1.48607$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 + 0.382i)3-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)9-s + (0.923 − 0.382i)11-s + (−0.382 + 0.923i)13-s i·17-s + (−0.382 + 0.923i)19-s + (−0.923 + 0.382i)21-s + (0.707 + 0.707i)23-s + (0.382 + 0.923i)27-s + (0.923 + 0.382i)29-s − 31-s + 33-s + (0.382 + 0.923i)37-s + (−0.707 + 0.707i)39-s + ⋯
L(s)  = 1  + (0.923 + 0.382i)3-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)9-s + (0.923 − 0.382i)11-s + (−0.382 + 0.923i)13-s i·17-s + (−0.382 + 0.923i)19-s + (−0.923 + 0.382i)21-s + (0.707 + 0.707i)23-s + (0.382 + 0.923i)27-s + (0.923 + 0.382i)29-s − 31-s + 33-s + (0.382 + 0.923i)37-s + (−0.707 + 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.471 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.471 + 0.881i$
Analytic conductor: \(1.48607\)
Root analytic conductor: \(1.48607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 320,\ (0:\ ),\ 0.471 + 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.405110285 + 0.8421906946i\)
\(L(\frac12)\) \(\approx\) \(1.405110285 + 0.8421906946i\)
\(L(1)\) \(\approx\) \(1.307488098 + 0.3833988023i\)
\(L(1)\) \(\approx\) \(1.307488098 + 0.3833988023i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.382 + 0.923i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 - T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (0.923 - 0.382i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.923 + 0.382i)T \)
59 \( 1 + (0.382 + 0.923i)T \)
61 \( 1 + (-0.923 - 0.382i)T \)
67 \( 1 + (0.923 + 0.382i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (0.382 - 0.923i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.15121979019150812851990450830, −24.20820283503842120908485464069, −23.2852263784638790192647848174, −22.359025884950717067129819499169, −21.34805087438800797187431144050, −20.15306159206075010676375212221, −19.7515010874929556693778435438, −19.0180252386170409919878187333, −17.72074582698721516779972091302, −17.02393006953814676783419244146, −15.74719269653693932435312557657, −14.8369517062626667623384489989, −14.11662653032791246306274179718, −12.84106903606136781347873854138, −12.67048713329860308972401917865, −10.98871982320731574384961378444, −9.9350204786052885561580644558, −9.11122338504512121383408929882, −8.0475278014384324857316534183, −7.04085704246141390671609115094, −6.29484605372882401656448786984, −4.50465614210576701749787320724, −3.54068956722520816142097098850, −2.48791762377257227207825760244, −1.02725720815402650071384167414, 1.741683334614964726426153812617, 2.94128850662452654458552894798, 3.83898005744582268649950552167, 5.06253868842920260449302369096, 6.42216914449246975339937005480, 7.40843430993545705624079421211, 8.81217696028533518900506929693, 9.22013768803244464337363195541, 10.18118598126680996015122897525, 11.554996565263570023400557474616, 12.45414972336431763450797588805, 13.635895333043893069101355942, 14.35273952961020673449615328868, 15.25375134212572093895293839848, 16.196759978239763607219623885699, 16.88735715970243759866321323763, 18.48402916690847353741275030492, 19.11452392759554216292839748494, 19.82768900285788932447824361942, 20.86661572359776019502611706123, 21.76550692977121592988974572239, 22.31356993608543427365359777972, 23.575695033304393584112270403869, 24.78868349477786523344469090468, 25.24346595067859671898246617991

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