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Properties

Label 1-71-71.44-r1-0-0
Degree $1$
Conductor $71$
Sign $0.685 - 0.728i$
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  + (−0.393 − 0.919i)2-s + (0.983 − 0.178i)3-s + (−0.691 + 0.722i)4-s + (0.309 + 0.951i)5-s + (−0.550 − 0.834i)6-s + (−0.753 − 0.657i)7-s + (0.936 + 0.351i)8-s + (0.936 − 0.351i)9-s + (0.753 − 0.657i)10-s + (0.963 + 0.266i)11-s + (−0.550 + 0.834i)12-s + (0.963 − 0.266i)13-s + (−0.309 + 0.951i)14-s + (0.473 + 0.880i)15-s + (−0.0448 − 0.998i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.393 − 0.919i)2-s + (0.983 − 0.178i)3-s + (−0.691 + 0.722i)4-s + (0.309 + 0.951i)5-s + (−0.550 − 0.834i)6-s + (−0.753 − 0.657i)7-s + (0.936 + 0.351i)8-s + (0.936 − 0.351i)9-s + (0.753 − 0.657i)10-s + (0.963 + 0.266i)11-s + (−0.550 + 0.834i)12-s + (0.963 − 0.266i)13-s + (−0.309 + 0.951i)14-s + (0.473 + 0.880i)15-s + (−0.0448 − 0.998i)16-s + (0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.685 - 0.728i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ 0.685 - 0.728i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.733775456 - 0.7493756116i\)
\(L(\frac12)\) \(\approx\) \(1.733775456 - 0.7493756116i\)
\(L(1)\) \(\approx\) \(1.217621476 - 0.4241454393i\)
\(L(1)\) \(\approx\) \(1.217621476 - 0.4241454393i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.393 - 0.919i)T \)
3 \( 1 + (0.983 - 0.178i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.753 - 0.657i)T \)
11 \( 1 + (0.963 + 0.266i)T \)
13 \( 1 + (0.963 - 0.266i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (0.473 - 0.880i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
29 \( 1 + (0.134 - 0.990i)T \)
31 \( 1 + (0.0448 - 0.998i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
43 \( 1 + (-0.995 + 0.0896i)T \)
47 \( 1 + (-0.983 - 0.178i)T \)
53 \( 1 + (0.691 + 0.722i)T \)
59 \( 1 + (0.550 - 0.834i)T \)
61 \( 1 + (-0.753 + 0.657i)T \)
67 \( 1 + (0.691 - 0.722i)T \)
73 \( 1 + (-0.393 - 0.919i)T \)
79 \( 1 + (0.936 + 0.351i)T \)
83 \( 1 + (-0.550 + 0.834i)T \)
89 \( 1 + (-0.691 - 0.722i)T \)
97 \( 1 + (-0.623 - 0.781i)T \)
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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

−32.00483023254814087707630199496, −30.92270184437528687269349029012, −29.15789145275046550726632414750, −28.00487568748184214485823694858, −27.12913840226168444400292940949, −25.79533918399192605550118089791, −25.11685940472632512190922137001, −24.486289738335163592330327553661, −22.970585706099606792012715129972, −21.5983220542857799713441643034, −20.31467621597425990069048458252, −19.2018137141241824194435690860, −18.25591007978900725674882377197, −16.43545390822383802777462010247, −16.10599843541637979476737810776, −14.58138772627825367511929541617, −13.6778496272499237135683455267, −12.4321531717444832525725621841, −10.02004254553482880650879913395, −9.02149860884905454164298116855, −8.442325887333057206630294739819, −6.72423526132497409047079092267, −5.33225983812267939841977207720, −3.688103098896539905475606725186, −1.3618824236633760159260767059, 1.39077998688152601291444563196, 3.05925551691203155103701546348, 3.83649750901233818268726674683, 6.64508267827992591527278623842, 7.896873567067818747253825790461, 9.426264371181428558087856203098, 10.139639514402934924570134886545, 11.56219129012517583379032635412, 13.21900070492530874542707594897, 13.811869070021274481091895021053, 15.219108926260826454226350626373, 17.00674049485770947517140165434, 18.259783165416585472667903505785, 19.24347382799215445787587116176, 19.95852224245806201539999258455, 21.14149819585589439635790290575, 22.24298781183955282763417815799, 23.31474712368851291225985778968, 25.34324428344882879208508942157, 25.97453210709417286072995249884, 26.796731244471122451967578485405, 28.01340228276252563380859373405, 29.54198620527598961766104374807, 30.16474162155151507585725314715, 30.82674762056084427944650477030

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