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Properties

Label 1-160-160.117-r1-0-0
Degree $1$
Conductor $160$
Sign $0.681 - 0.731i$
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 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.681 - 0.731i$
Analytic conductor: \(17.1943\)
Root analytic conductor: \(17.1943\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 160,\ (1:\ ),\ 0.681 - 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.115949172 - 0.4856051922i\)
\(L(\frac12)\) \(\approx\) \(1.115949172 - 0.4856051922i\)
\(L(1)\) \(\approx\) \(0.8422766317 - 0.1754719510i\)
\(L(1)\) \(\approx\) \(0.8422766317 - 0.1754719510i\)

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

−27.83195525198550696163545926709, −26.7449404236772707270584139574, −25.98012476140491523268441605939, −24.79983189060782144406689881457, −23.60322928732066860921989119781, −22.61801108747002156897099071409, −22.08451781373504026726841285418, −20.97171753139354403738672663537, −19.90623072040071038284313842111, −18.80329895176410266731239501059, −17.63802675067153221687752334154, −16.59228618827960800283443575789, −15.95149804787243933375480096436, −14.90234826955453397165486596438, −13.53791404171064591342090566805, −12.38737354314522257536152801332, −11.364883453870648733982502707937, −10.2630884121016177488882037533, −9.45018314202338024561102820858, −8.118404788203705450433413397136, −6.25991340408384533918838973340, −5.88712151576391223592351498351, −4.118193115025672716042126791832, −3.25753194418786570519622725135, −0.90810064241052799065378924246, 0.70971676612110625995111837064, 2.26227731571864012450127687194, 3.96482913978465783718288693727, 5.4523437183075637679474471298, 6.63133713562121059970851551231, 7.26980064439488763239418283870, 8.9570018044254935231063752469, 10.0209713742836192582605107700, 11.42313325213142301057193349258, 12.15721848590638007891682293837, 13.258398227949049256618709380084, 14.10377907892873725419803687268, 15.80036786401894708920110949609, 16.48193073874636508109260955550, 17.63115067048537942979055217849, 18.49848574959288570503295770778, 19.45316169667034029703230723349, 20.38783565283618083126895554544, 21.98658466621505004203363803304, 22.61162142689524535025186599996, 23.46465166720047397967414141163, 24.50772114545994777673729712145, 25.42564627468460115053659611454, 26.3248997109346932465375789965, 27.77528022433690495039048996447

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