Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{2} $
Sign $0.744 + 0.667i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.891 − 0.453i)3-s + 7-s + (0.587 − 0.809i)9-s + (−0.987 − 0.156i)11-s + (0.156 + 0.987i)13-s + (−0.951 + 0.309i)17-s + (0.891 + 0.453i)19-s + (0.891 − 0.453i)21-s + (−0.809 + 0.587i)23-s + (0.156 − 0.987i)27-s + (0.453 + 0.891i)29-s + (0.309 + 0.951i)31-s + (−0.951 + 0.309i)33-s + (0.987 − 0.156i)37-s + (0.587 + 0.809i)39-s + ⋯
L(s,χ)  = 1  + (0.891 − 0.453i)3-s + 7-s + (0.587 − 0.809i)9-s + (−0.987 − 0.156i)11-s + (0.156 + 0.987i)13-s + (−0.951 + 0.309i)17-s + (0.891 + 0.453i)19-s + (0.891 − 0.453i)21-s + (−0.809 + 0.587i)23-s + (0.156 − 0.987i)27-s + (0.453 + 0.891i)29-s + (0.309 + 0.951i)31-s + (−0.951 + 0.309i)33-s + (0.987 − 0.156i)37-s + (0.587 + 0.809i)39-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.744 + 0.667i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.744 + 0.667i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(800\)    =    \(2^{5} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.744 + 0.667i$
motivic weight  =  \(0\)
character  :  $\chi_{800} (533, \cdot )$
Sato-Tate  :  $\mu(40)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 800,\ (1:\ ),\ 0.744 + 0.667i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.779622168 + 1.062824417i$
$L(\frac12,\chi)$  $\approx$  $2.779622168 + 1.062824417i$
$L(\chi,1)$  $\approx$  1.539244167 + 0.02224977156i
$L(1,\chi)$  $\approx$  1.539244167 + 0.02224977156i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−21.89620477289199725726514510278, −20.96420867191087258211987462829, −20.37481635439153229349204348541, −19.95085373252500652380569179087, −18.62765103251731170185689417263, −18.11298002691151323805994367250, −17.236339214419154258892406561124, −15.99989100007320812198893841416, −15.4096105676268541849603716305, −14.84549529971337053689089061652, −13.64860569065856552860835418412, −13.425608269197622245699537931318, −12.13089135730180084668469632620, −11.08477218422864275010485333662, −10.35916769224557911467732183945, −9.56162253479034635164122932929, −8.39375691441588888685304113655, −8.02067685694950451836145920579, −7.11395140818563367104178997074, −5.59618448539592272522887910673, −4.81696206409344750132869503504, −3.97818841015386964844074813707, −2.7098092226302024852384369325, −2.13012464285033815204834295350, −0.58278082098890332233425538654, 1.207162200809761241283888234632, 1.99444388769109207850301829542, 2.98918144636219272564182525287, 4.11479297864952907775732540614, 5.0078112690552756376374950212, 6.23266801093906783477491925982, 7.25191968857865785256260945042, 8.0094706500013015824966652826, 8.65478964833107416515338808020, 9.576528683395517464744594111360, 10.61681038032836471613194155213, 11.56392055579485819855446426745, 12.370333842280459272356306470935, 13.45474114715657856659284508951, 13.94250733191575240618234699865, 14.72537717976023256050645597608, 15.594234229754735816875956627668, 16.35656866000149279179169535584, 17.70256751408361178004200783205, 18.182095378742104454154302556949, 18.85698605696585880552913906537, 19.970251380383999622236575861968, 20.36421216393838720549050973078, 21.468074580959249330005244473955, 21.69225203949531468210682777345

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