Properties

Label 1-160-160.99-r1-0-0
Degree $1$
Conductor $160$
Sign $0.195 + 0.980i$
Analytic cond. $17.1943$
Root an. cond. $17.1943$
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 i·7-s + i·9-s + (−0.707 + 0.707i)11-s + (0.707 + 0.707i)13-s + 17-s + (0.707 + 0.707i)19-s + (0.707 − 0.707i)21-s i·23-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)29-s − 31-s − 33-s + (0.707 − 0.707i)37-s + i·39-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s i·7-s + i·9-s + (−0.707 + 0.707i)11-s + (0.707 + 0.707i)13-s + 17-s + (0.707 + 0.707i)19-s + (0.707 − 0.707i)21-s i·23-s + (−0.707 + 0.707i)27-s + (0.707 + 0.707i)29-s − 31-s − 33-s + (0.707 − 0.707i)37-s + i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.195 + 0.980i$
Analytic conductor: \(17.1943\)
Root analytic conductor: \(17.1943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 160,\ (1:\ ),\ 0.195 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.721003616 + 1.412391167i\)
\(L(\frac12)\) \(\approx\) \(1.721003616 + 1.412391167i\)
\(L(1)\) \(\approx\) \(1.313215722 + 0.4550290505i\)
\(L(1)\) \(\approx\) \(1.313215722 + 0.4550290505i\)

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 \( 1 - iT \)
11 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 - iT \)
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 \( 1 - T \)
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 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 - iT \)
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

−27.3102614756079199059072128417, −26.138918650689403210837787473319, −25.40886162961021228566574197623, −24.5351439090317000152605500536, −23.70777036425716236317860571544, −22.537749064277072599064302636790, −21.28051894062562836421269826609, −20.534164466419081337158458286466, −19.31136538124523697849141465224, −18.50447565258377279466919319316, −17.877830751073253262401810102910, −16.20298010150748552096198555004, −15.28829556018160608769172132256, −14.24225902411754654794039587117, −13.189329032636127451374180710973, −12.3726560423417732801726110301, −11.18623183474497870266554898397, −9.692857050835638580935234667894, −8.52105354918853032455534156627, −7.87026849512033578737663159179, −6.370587339601507198135591903965, −5.34604628045226149558214638224, −3.3523197927803615008056170253, −2.47669146401575743461384319162, −0.81279357542506030210571043235, 1.546138939193045621902819053934, 3.23687435219309002675941220262, 4.185747409478120867595123533336, 5.42473419453492462739379603603, 7.228708420957769215119932018305, 8.05433001800731911970480745122, 9.46838668072146138446266191731, 10.21964028013640399849438498361, 11.2708675915975810480110624504, 12.85563549241718225151265643635, 13.915781705006538805071443428880, 14.62138720909575535957293710829, 15.95889986427940455883133892612, 16.53401501325985684532733172608, 17.91499653218612981354248110422, 19.09057778890297249980196896558, 20.17736122340070537252129846181, 20.80073005029934349291137390404, 21.69527847029367336631254174971, 23.0560860675975545070608603993, 23.696743167849485529197420980096, 25.25000397570920190025005998799, 25.88369664254338590009533897806, 26.7570742618819118498018917295, 27.59247506715170332627888051941

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