Properties

Label 1-160-160.27-r0-0-0
Degree $1$
Conductor $160$
Sign $0.681 - 0.731i$
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

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

Invariants

Degree: \(1\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.681 - 0.731i$
Analytic conductor: \(0.743036\)
Root analytic conductor: \(0.743036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 160,\ (0:\ ),\ 0.681 - 0.731i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9101689850 - 0.3960599601i\)
\(L(\frac12)\) \(\approx\) \(0.9101689850 - 0.3960599601i\)
\(L(1)\) \(\approx\) \(0.9307089166 - 0.2308418525i\)
\(L(1)\) \(\approx\) \(0.9307089166 - 0.2308418525i\)

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

−27.76812729835312519120152392208, −27.09916483923375163040190591859, −26.37816417484005399250202489286, −24.82352256882932393682465446086, −24.011139996471189872301310820550, −23.07495555592713017557178153200, −21.808224933879595074275047237810, −21.44590729694975573710082786102, −20.2676559661811969639506066564, −19.02905933360468029958465077724, −17.808884617789974818133173180772, −16.9798950088707752585489789392, −16.203469765259166250551083422899, −14.81401985251042175900980218141, −14.25914144294554027685091715853, −12.49998516235453340363437410930, −11.51356235816057314164650727801, −10.78943558935228141898574537798, −9.5251859056442567911175100783, −8.46664669199954949379154105112, −6.94973935488019096893675409397, −5.685494188128314379734287038845, −4.67485417028504886219423317308, −3.530773444132406594607545375852, −1.45868257978250108641184873182, 1.12370116796876730111395763647, 2.5402723979605657848810833565, 4.64167992884722649442906956267, 5.43179179565065155000769772192, 7.00101161715341240687292288546, 7.62939125895061455810439256511, 9.117904911451569125756618882871, 10.54085357515506456937353340585, 11.60601705691310793566410543032, 12.28978246799140132826021877584, 13.54557928310559914778068074063, 14.5642152528688991566300065510, 15.751854811414704242879619450066, 17.247531760160840037635618297186, 17.56902225142179235751192314212, 18.63798738165399379893945393341, 19.79550064766797918332238355053, 20.756247126382788245761102148007, 22.15778004891226362550678544665, 22.75478013566759636648183901874, 23.92237206584886463439306979317, 24.677101564265189855952110412200, 25.40249019661867909653934988382, 27.130680368759292004122128828790, 27.59533133494779176717527218787

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