Properties

Degree 1
Conductor $ 2^{2} \cdot 5^{2} $
Sign $0.998 + 0.0627i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.998 + 0.0627i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.998 + 0.0627i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(100\)    =    \(2^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $0.998 + 0.0627i$
motivic weight  =  \(0\)
character  :  $\chi_{100} (67, \cdot )$
Sato-Tate  :  $\mu(20)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 100,\ (0:\ ),\ 0.998 + 0.0627i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.344010530 + 0.04223723248i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.344010530 + 0.04223723248i\)
\(L(\chi,1)\)  \(\approx\)  \(1.332519553 + 0.03963542506i\)
\(L(1,\chi)\)  \(\approx\)  \(1.332519553 + 0.03963542506i\)

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

−30.21787230497937245022156172844, −28.96838168445643838330924738523, −27.77885428002611447113362234848, −26.79957802453597471593944286939, −25.5507053556061478000642747737, −24.926905513336786189766507102135, −24.04425117619190345683261722894, −22.43299828841005229064069633119, −21.60956514299968542264095553538, −20.17833602930944671870716442538, −19.618994905522756081943140306782, −18.33403930378635978831524393316, −17.47084500326566153824703174831, −15.57927278528974942963367523898, −15.04754098801139674736431781707, −13.77418575163566803571514704779, −12.66072538095144967204631890456, −11.63430814477296189633952558976, −9.75801724716987796765338642782, −8.9599053431575930350993413611, −7.72496915799651067854093639661, −6.486745264046017039204599190286, −4.78122236343968103402597778414, −3.12731275590550216267690687717, −1.92875516039207501279766894184, 1.827673324264072119471936429497, 3.574781326000960575390329380239, 4.48073852212963786376546929943, 6.54768160312709283897155587515, 7.755010774003055668450372642186, 8.9737294379066012579957777209, 10.02505014427654943409177181004, 11.24464802671502252890933414243, 12.85664163643860368610602909188, 14.06304879518695349567600644838, 14.59503147731974968971500435768, 16.16486266566674886706415387900, 16.9431474574376889082272391715, 18.55657371369434894647881062753, 19.67721791941591693799553070508, 20.30457380606400699848308741534, 21.52481445073532608300723572087, 22.46651147883687013780326049324, 24.02421035999960779441560374367, 24.70791683207060199967017404443, 26.09277140790248426627748321358, 26.712066529658055408026095775059, 27.56497246349771649123732700575, 29.12741853549087757811313216044, 30.05041653627839790949167955348

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