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Properties

Label 1-71-71.26-r1-0-0
Degree $1$
Conductor $71$
Sign $-0.981 - 0.189i$
Analytic cond. $7.63000$
Root an. cond. $7.63000$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $-0.981 - 0.189i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ -0.981 - 0.189i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1972070546 - 2.058781326i\)
\(L(\frac12)\) \(\approx\) \(0.1972070546 - 2.058781326i\)
\(L(1)\) \(\approx\) \(0.8680580018 - 1.166397078i\)
\(L(1)\) \(\approx\) \(0.8680580018 - 1.166397078i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.623 - 0.781i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + T \)
7 \( 1 + (-0.623 - 0.781i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (0.900 + 0.433i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.222 + 0.974i)T \)
23 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 + (0.900 + 0.433i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (0.900 + 0.433i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (0.222 - 0.974i)T \)
59 \( 1 + (0.900 - 0.433i)T \)
61 \( 1 + (-0.623 + 0.781i)T \)
67 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 + (-0.900 - 0.433i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (0.900 - 0.433i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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