Properties

Label 1-160-160.67-r0-0-0
Degree $1$
Conductor $160$
Sign $-0.936 - 0.349i$
Analytic cond. $0.743036$
Root an. cond. $0.743036$
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  + (−0.707 + 0.707i)3-s − 7-s i·9-s + (−0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s i·17-s + (−0.707 − 0.707i)19-s + (0.707 − 0.707i)21-s − 23-s + (0.707 + 0.707i)27-s + (−0.707 − 0.707i)29-s − 31-s i·33-s + (−0.707 − 0.707i)37-s i·39-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s − 7-s i·9-s + (−0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s i·17-s + (−0.707 − 0.707i)19-s + (0.707 − 0.707i)21-s − 23-s + (0.707 + 0.707i)27-s + (−0.707 − 0.707i)29-s − 31-s i·33-s + (−0.707 − 0.707i)37-s i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.936 - 0.349i$
Analytic conductor: \(0.743036\)
Root analytic conductor: \(0.743036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 160,\ (0:\ ),\ -0.936 - 0.349i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01434672179 + 0.07946721892i\)
\(L(\frac12)\) \(\approx\) \(0.01434672179 + 0.07946721892i\)
\(L(1)\) \(\approx\) \(0.5014135764 + 0.1289956140i\)
\(L(1)\) \(\approx\) \(0.5014135764 + 0.1289956140i\)

Euler product

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

−27.47154262855507806149596151162, −26.15393923141976460143802814856, −25.32685432989472070920025925557, −24.1611641306812201323104026169, −23.50418879580785515265623559366, −22.386656611695262754941493692180, −21.78039410125735078934945152253, −20.20762520010357395382381142367, −19.16551957014875198121345168586, −18.51958249350204016684345916859, −17.27764743248213024606588745724, −16.50669449583835489936493332916, −15.45633446569099604976603066406, −13.979992082213404143461251720113, −12.807407314757844309011800198066, −12.40303200141454885105371420132, −10.85549560130987711471079101280, −10.12284789960212253587844724852, −8.450876832163183792146529506562, −7.37616670755221132499709783595, −6.16598330497692058024087981875, −5.39680002967222887571845194176, −3.577243813047840912328796192228, −2.06577950341454943502600405157, −0.06768012176402300070191595438, 2.50518602063757181244917705505, 4.028856621838587668814407844805, 5.08328056982625538899133901446, 6.326651494816406864694177100602, 7.37701641321528120128902462558, 9.27952198158727838087543058656, 9.838666395608220720590379381401, 11.008736904093074279398367787, 12.13494675736584108175121744426, 13.04734825024154412931685622188, 14.52471741608865473592954381627, 15.66334233573805815243740378791, 16.30967889735451683733424441620, 17.35322396707411441554908286810, 18.36052215616774458762247013461, 19.58774954107345883350483116789, 20.64023080795777901877468725426, 21.67740705465832644605225980619, 22.51499913092603884451430381762, 23.2747794101190504299326375357, 24.303577434440594904215867180261, 25.818846988146875957874074930932, 26.32462013298385202829092108049, 27.48288093400341670954479876371, 28.44142693178944696109685343942

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