Properties

Label 1-160-160.29-r0-0-0
Degree $1$
Conductor $160$
Sign $0.980 - 0.195i$
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 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

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

Invariants

Degree: \(1\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.980 - 0.195i$
Analytic conductor: \(0.743036\)
Root analytic conductor: \(0.743036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 160,\ (0:\ ),\ 0.980 - 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9399043588 - 0.09257249932i\)
\(L(\frac12)\) \(\approx\) \(0.9399043588 - 0.09257249932i\)
\(L(1)\) \(\approx\) \(0.9159359761 - 0.08870271340i\)
\(L(1)\) \(\approx\) \(0.9159359761 - 0.08870271340i\)

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

−27.62655678462834963101445487728, −27.218464074880442074902892710244, −25.964443327486103962853157270997, −25.104887426051066969484725101090, −23.35157033708730213600637486404, −23.249892408060797930569911495804, −22.08864627850175523489303325478, −20.91380996009915756414285771755, −20.29739629253567930639714366253, −18.974293412230138013031269111300, −17.51091809850142624804759200320, −17.09860636748353433086098517956, −15.9605131401905760929222470641, −14.980322132904965143140208225491, −13.85557918963202567699094845939, −12.51166790085775815861318330670, −11.49806483870633247463091205525, −10.34865782975098432837169526844, −9.75980819920496070984847058472, −8.16857683504310978184426401623, −6.81291191482844248921578392381, −5.713782118108967505762248502471, −4.37265790609157842044250039752, −3.50604072904881962693291245826, −1.159236424146795280751022833880, 1.275182040258837704516333548320, 2.77204709067994771874893669867, 4.59670553973944249363404080637, 5.97068946534162999696444424555, 6.57281681814134194530001272287, 8.14199992255466671571466989346, 9.08255607396122149413503427503, 10.70863235490435589723958806748, 11.68363271462946743860346199674, 12.420653097504642539977100671257, 13.59507243067272087575549348135, 14.66750380393319300679113608461, 16.10562710351832321678507838234, 16.82411401119268390247918875534, 18.063444235904352880631226470094, 18.81020390024137075085845374792, 19.56346681230996411796203680406, 21.26493468860377700608180291585, 21.91608816926790102258969184394, 23.02538870819644928162011938496, 23.897727722430768152482114029919, 24.83381023085004044044040507002, 25.55667036374589463479484814659, 26.96645962812888362421114757400, 28.18380708744235394868965813459

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