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Properties

Label 1-320-320.163-r0-0-0
Degree $1$
Conductor $320$
Sign $0.502 - 0.864i$
Analytic cond. $1.48607$
Root an. cond. $1.48607$
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  + (0.382 − 0.923i)3-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)11-s + (−0.923 − 0.382i)13-s + 17-s + (0.382 − 0.923i)19-s + (0.923 − 0.382i)21-s + (−0.707 + 0.707i)23-s + (−0.923 + 0.382i)27-s + (0.923 + 0.382i)29-s + 31-s − i·33-s + (0.923 − 0.382i)37-s + (−0.707 + 0.707i)39-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)11-s + (−0.923 − 0.382i)13-s + 17-s + (0.382 − 0.923i)19-s + (0.923 − 0.382i)21-s + (−0.707 + 0.707i)23-s + (−0.923 + 0.382i)27-s + (0.923 + 0.382i)29-s + 31-s − i·33-s + (0.923 − 0.382i)37-s + (−0.707 + 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.502 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.502 - 0.864i$
Analytic conductor: \(1.48607\)
Root analytic conductor: \(1.48607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 320,\ (0:\ ),\ 0.502 - 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.341187111 - 0.7718784346i\)
\(L(\frac12)\) \(\approx\) \(1.341187111 - 0.7718784346i\)
\(L(1)\) \(\approx\) \(1.222773782 - 0.3957375508i\)
\(L(1)\) \(\approx\) \(1.222773782 - 0.3957375508i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + (-0.923 - 0.382i)T \)
17 \( 1 + T \)
19 \( 1 + (0.382 - 0.923i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + T \)
37 \( 1 + (0.923 - 0.382i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.382 - 0.923i)T \)
59 \( 1 + (-0.382 - 0.923i)T \)
61 \( 1 + (0.923 + 0.382i)T \)
67 \( 1 + (-0.382 + 0.923i)T \)
71 \( 1 + (-0.707 + 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 - iT \)
83 \( 1 + (-0.923 - 0.382i)T \)
89 \( 1 + (0.707 - 0.707i)T \)
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

−25.24098761730711866508700041887, −24.612047485665810188319938166550, −23.36416156068464096216263061090, −22.55512660946350014809884905156, −21.61455349172934222328204952391, −20.83619490219674790910071202877, −20.046161138103404450336819511142, −19.31211958442107541614955732307, −17.962967796915039206825603085756, −16.843201670844973496725315930866, −16.51183463421035840310938915262, −15.06898161379644785558607707159, −14.40600820480397632481006853278, −13.85452470565394482908718817329, −12.19415712658366393642765048891, −11.451848028481049347239994037583, −10.09019472541593600061076744815, −9.79891556419205726603802470797, −8.373350821696605845486654757351, −7.62986235121345827587350577110, −6.24077926015276959500481228492, −4.79257295711664105703432905208, −4.233411269643086138174614550189, −2.99744756273353747736934218279, −1.52908123210687881398740906993, 1.113984179627944206938361817216, 2.31028971982916703189713938527, 3.367134509112551644046440315573, 4.99034612689518498176341486643, 6.01642565442474814492552840063, 7.138123164952357570792505999542, 8.068945875123677843743122366479, 8.88990405840519465103323548759, 9.9624284325093575505014223939, 11.67864581654090098944599764245, 11.89224610288423146398768906277, 13.07205178479528218420955579426, 14.20168949043887266716414102910, 14.66221110467475081409098962130, 15.8174961244379055783310269568, 17.26123157838879182675503129742, 17.739899672466726310794738740490, 18.81478339073363062935778622400, 19.5230150573083435565017885025, 20.33332238202977078304618751065, 21.51148708163600595303352358950, 22.24166352248674631869830676279, 23.48066714876367336490148139613, 24.24642139462696979536606442645, 24.940622801150878425089644686059

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