↑

Properties

Label 1-320-320.203-r0-0-0
Degree $1$
Conductor $320$
Sign $0.998 - 0.0626i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $0.998 - 0.0626i$
Analytic conductor: \(1.48607\)
Root analytic conductor: \(1.48607\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{320} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 320,\ (0:\ ),\ 0.998 - 0.0626i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.046297951 - 0.03281245637i\)
\(L(\frac12)\) \(\approx\) \(1.046297951 - 0.03281245637i\)
\(L(1)\) \(\approx\) \(0.9331562610 - 0.1035312902i\)
\(L(1)\) \(\approx\) \(0.9331562610 - 0.1035312902i\)

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 \( 1 - iT \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.29749220075181796681004424996, −24.12352731009565525919349437356, −23.166602769091183170283748746364, −22.443075296275043552830788843364, −21.7433381756688799942777801169, −20.680290344866371334007493180467, −19.938050735585017689351833907788, −19.01622964443971032708205386607, −17.64375158821417939910256607285, −16.963899161321301460543788823018, −16.06057980403738551133401208641, −15.45270112067304909587551066261, −14.14728015955208359661749760433, −13.43646357527926914542171037765, −12.07924310122940147314050333092, −11.10221172815290142721868032269, −10.43562334899475737321869146142, −9.25069584584799095472892633322, −8.67025077680851030126233893428, −6.800019565007208665168732139834, −6.29835855212152457259544771963, −4.802292141969901242266106824201, −3.953800958787982998647400248775, −2.97804748157707332203053181756, −0.87864076698793895052247800359, 1.18073078826680998515456254301, 2.42905553535249540755091465648, 3.71107621286638732950188668999, 5.303453302199632265766826886731, 6.273923309341860751173673338499, 6.920809037526689788748807858005, 8.26316148267079599014612018332, 9.09620977713430494682676201670, 10.366249478905310555014147315416, 11.59248924866322684530699292286, 12.20068086639482796612438244734, 13.17919237154329703661840709878, 13.92651731518454978653518172326, 15.24024354912551554761884507648, 16.09729499748518071706361978840, 17.22750990077173182546870296794, 17.93282379205257367592564916535, 18.902956737382462885213059903338, 19.51722676459186763529741196397, 20.52636053373135938269928992597, 21.816535347843355273164493438792, 22.709758380315831791662454499038, 23.13809601161513405482537003400, 24.411065233773130107620756740808, 25.132823711987017139585718575952

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