Properties

Degree 1
Conductor $ 2^{2} \cdot 5^{2} $
Sign $0.425 + 0.904i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.425 + 0.904i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.425 + 0.904i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(100\)    =    \(2^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.425 + 0.904i$
motivic weight  =  \(0\)
character  :  $\chi_{100} (59, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 100,\ (1:\ ),\ 0.425 + 0.904i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.152259700 + 0.7312462417i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.152259700 + 0.7312462417i\)
\(L(\chi,1)\)  \(\approx\)  \(0.9358598968 + 0.2540250827i\)
\(L(1,\chi)\)  \(\approx\)  \(0.9358598968 + 0.2540250827i\)

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

−29.648336246280855743902313948205, −28.29389592309771258936881911290, −27.79284901168348543510186134930, −26.51888450800937232553276592063, −24.97156220988477854066874124064, −24.41251454024979104337849073111, −23.14420343092907230534774892912, −22.51269811540296234566486856716, −21.09606106817307240756140497399, −20.08890603279608935027598439530, −18.592914512003524128651541180257, −17.8323258625520663390390987578, −17.01505058877383710544906544375, −15.61894319877056381564962476881, −14.40144086701674480739211980527, −13.05380073458289411470742942131, −12.05884164505776065298772794622, −11.05339551512235533069173483399, −9.867910571362079489000546002057, −7.975732734127437543419132903096, −7.2124093265236748327092897608, −5.568335664713866196274496847317, −4.690024050922154656117754448364, −2.38876140162935409427990490726, −0.792213449768409302693341609757, 1.298207143180879839028718267543, 3.555418614831584361195134171829, 4.94659941145312019039306062752, 5.903977872380190143192313822978, 7.51536704981603626340804836678, 8.949389890869736027196310587588, 10.28633586044067894083763015481, 11.327677058938488326085917184956, 12.127258559032661927367061469768, 13.850566925811140406548961270076, 14.90968176023100092777196599135, 16.17547052947023803600766315794, 17.00851786843568440386897379672, 18.07547485728295299584238691502, 19.17987175836405276145918954940, 20.93087518459753714318474474348, 21.34887291688026555442951841509, 22.500746527933434973022986903622, 23.77168627725284455223865792646, 24.288622393715059497473679419531, 25.99040166984662601954589267086, 27.03945801957379322856626174956, 27.67254527094758985097710054763, 28.862864317648190569354303924637, 29.62348675175455948657143170016

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