Properties

Degree 1
Conductor 71
Sign $-0.981 + 0.189i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)3-s + (−0.222 + 0.974i)4-s + 5-s + (−0.900 + 0.433i)6-s + (−0.623 + 0.781i)7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.623 + 0.781i)10-s + (0.900 + 0.433i)11-s + (−0.900 − 0.433i)12-s + (0.900 − 0.433i)13-s − 14-s + (−0.222 + 0.974i)15-s + (−0.900 − 0.433i)16-s − 17-s + ⋯
L(s,χ)  = 1  + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)3-s + (−0.222 + 0.974i)4-s + 5-s + (−0.900 + 0.433i)6-s + (−0.623 + 0.781i)7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.623 + 0.781i)10-s + (0.900 + 0.433i)11-s + (−0.900 − 0.433i)12-s + (0.900 − 0.433i)13-s − 14-s + (−0.222 + 0.974i)15-s + (−0.900 − 0.433i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.981 + 0.189i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.981 + 0.189i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $-0.981 + 0.189i$
motivic weight  =  \(0\)
character  :  $\chi_{71} (41, \cdot )$
Sato-Tate  :  $\mu(14)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 71,\ (1:\ ),\ -0.981 + 0.189i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.1972070546 + 2.058781326i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.1972070546 + 2.058781326i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8680580018 + 1.166397078i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8680580018 + 1.166397078i\)

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

−30.4740242763657227437531673779, −29.92308515228040949683894683147, −29.015292657370460377135005141128, −28.32377017585941479931105704111, −26.54490090458320537503248442527, −25.126866738197530284180258451150, −24.239982408160402939288003434726, −23.012274175984549121579701723769, −22.284698681190603936040218653861, −20.93243336657778955273672735076, −19.801334330786671728133255921976, −18.828182113271504894135713802697, −17.70983923732700927504599300413, −16.458211029540854083795298650254, −14.26542402086377187189722919101, −13.63631743645000416551508757704, −12.758014651258259185221834201244, −11.42732699456737750319393664853, −10.266102221558276911629345400275, −8.84870390347033503388908561339, −6.56827645825172395877868623127, −5.99969209107435885876441727429, −3.992118952082728207695929834338, −2.24052602379609027195986771425, −0.9194916361959866971185707092, 2.844479329795411345057359419709, 4.3758449589365584601916384826, 5.74564247540401083367342405773, 6.49588263293365142053827598418, 8.7952543603818659182047614777, 9.52393969868064278083580892310, 11.27635635805190107829123624830, 12.74502041795833781680457491004, 13.919834339383711748589701622590, 15.17381890090065741468638378772, 15.93434151801702910859686607087, 17.19131101846306494488837971181, 17.982803881713327358833387238, 20.12757335214581586618642101880, 21.411497465816845825930581994862, 22.10590377659126537315854049693, 22.80765097998088281373202812608, 24.41876948537222584514565949669, 25.649953464503600901082490365679, 25.96726370133662019757184625228, 27.57449143018824730891455318454, 28.56370448661153930550089469816, 29.89660991666521248611187931775, 31.19446533344730640453923202651, 32.39272901250379606138549795266

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