Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{2} $
Sign $-0.331 - 0.943i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.331 - 0.943i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.331 - 0.943i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(800\)    =    \(2^{5} \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.331 - 0.943i$
motivic weight  =  \(0\)
character  :  $\chi_{800} (53, \cdot )$
Sato-Tate  :  $\mu(40)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 800,\ (1:\ ),\ -0.331 - 0.943i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5012465172 - 0.7072812681i$
$L(\frac12,\chi)$  $\approx$  $0.5012465172 - 0.7072812681i$
$L(\chi,1)$  $\approx$  0.9108385591 + 0.05719404173i
$L(1,\chi)$  $\approx$  0.9108385591 + 0.05719404173i

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

−22.54754586277045082358048528593, −21.44545329089473122035916985419, −20.64396556244202150166534334715, −19.99453090580434616825427089644, −18.77829227147491076338926803627, −18.32653300844950659152938038059, −17.604649731931485016703693994559, −16.88741421883156010958874598137, −15.941219223659816309962964437794, −14.862081920927036778811957822523, −14.115779820094467431086843840328, −13.3283458073925660239256943768, −12.31146691410359832573918998825, −11.84824755682035890183369876673, −10.858931718434913685477252467988, −10.12829617551836292122561363503, −8.76507973922200777376215962127, −7.88628252796592682606200530039, −7.381562112432614679130887640520, −6.21137883679708955411039722985, −5.444357137492493347377547265, −4.53893228597987205787388448858, −3.24784295355413052664732105988, −1.77585937282117850275237608216, −1.40621860291298101492467851149, 0.204854458893208948390170624221, 1.398224339583099140442552495553, 2.87053574436634245705274371348, 3.92769202293263607587266932562, 4.65110765663722756076012177823, 5.7514611505349440333586600369, 6.22954215914019453388298505031, 7.813254731417123063247295736989, 8.49404116829390949505371096132, 9.38281271109282391268052905595, 10.4162753910576644686311978283, 11.22397926269091471340599638565, 11.51956431962238577384660927790, 12.80032839240049944254979996160, 13.85346733061549478359995804891, 14.62167299588287043993546421318, 15.35425291542507199803293780831, 16.311831851168090302600579183880, 16.793994711193178247043382164061, 17.79915382952643529155506324932, 18.4264775193667838450853560466, 19.45841151359025526998463581359, 20.56909503089147935560756382329, 21.131151621088075292873489276065, 21.62585071230769134080247702466

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