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Properties

Label 1-320-320.243-r0-0-0
Degree $1$
Conductor $320$
Sign $0.966 - 0.256i$
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  + (−0.923 − 0.382i)3-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)9-s + (0.382 + 0.923i)11-s + (−0.382 + 0.923i)13-s + 17-s + (−0.923 − 0.382i)19-s + (0.382 + 0.923i)21-s + (0.707 − 0.707i)23-s + (−0.382 − 0.923i)27-s + (0.382 − 0.923i)29-s + 31-s − i·33-s + (0.382 + 0.923i)37-s + (0.707 − 0.707i)39-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)3-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)9-s + (0.382 + 0.923i)11-s + (−0.382 + 0.923i)13-s + 17-s + (−0.923 − 0.382i)19-s + (0.382 + 0.923i)21-s + (0.707 − 0.707i)23-s + (−0.382 − 0.923i)27-s + (0.382 − 0.923i)29-s + 31-s − i·33-s + (0.382 + 0.923i)37-s + (0.707 − 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(1.48607\)
Root analytic conductor: \(1.48607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 320,\ (0:\ ),\ 0.966 - 0.256i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8468325872 - 0.1103035463i\)
\(L(\frac12)\) \(\approx\) \(0.8468325872 - 0.1103035463i\)
\(L(1)\) \(\approx\) \(0.7919480693 - 0.07953085166i\)
\(L(1)\) \(\approx\) \(0.7919480693 - 0.07953085166i\)

Euler product

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

−25.12582993925654118692512125947, −24.278352840368671863850930452312, −23.09740474696845950619279403174, −22.64694219399914827304628713214, −21.55887669111911311237840912009, −21.16758944590033829978938415641, −19.602329350046486821355829021219, −18.913725382416476793428066532400, −17.88643511406773593665994396371, −16.955242522383503766008148904795, −16.203555996433183108071298276204, −15.38433301603021206936568088078, −14.41513032560566235368278951485, −12.91889879098358645728250696902, −12.338323703005680969662028024311, −11.31768714114617610658167564318, −10.36552177045322234583464341572, −9.50042886089792929028802655199, −8.4141646839256803938678077439, −6.997186938945940727534794301849, −5.89602712244635431286601872029, −5.389436238892505923362643681299, −3.89737033445984048032318042788, −2.856328987446458398803592061624, −0.905209324548247176955956987415, 0.92925967199159908991313561883, 2.36864072743537765207379970629, 4.091426953552209682618936394462, 4.88283954710890837963124259364, 6.42146946195707366899112902757, 6.81955099032870575194516957477, 7.950089299536467194017996856, 9.564226572429243154544043000531, 10.22481727270406781063803644398, 11.34332578929681783242248329857, 12.2776790982316800732715751701, 12.98370587915747827637607352595, 14.03915403713884140678155684432, 15.17257416649648426627544946873, 16.4140819710312782924385076469, 16.95197323916586326953267605040, 17.698367331141944669171388297957, 19.01206398422249982639829261826, 19.3947643550950136567762588466, 20.73392051604620710673594578881, 21.66457711968042359002463220343, 22.79225730581298619868958420562, 23.117430717167911203628769066728, 24.05311836214224815390758118043, 25.04330968934659730353178266707

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