↑

Properties

Label 1-71-71.17-r1-0-0
Degree $1$
Conductor $71$
Sign $-0.230 - 0.973i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)5-s + (−0.809 + 0.587i)6-s + (−0.309 − 0.951i)7-s + (−0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)10-s + (−0.309 − 0.951i)11-s + (−0.809 − 0.587i)12-s + (−0.309 + 0.951i)13-s + (0.809 − 0.587i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)5-s + (−0.809 + 0.587i)6-s + (−0.309 − 0.951i)7-s + (−0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)10-s + (−0.309 − 0.951i)11-s + (−0.809 − 0.587i)12-s + (−0.309 + 0.951i)13-s + (0.809 − 0.587i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $-0.230 - 0.973i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ -0.230 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1497841610 + 0.1894426706i\)
\(L(\frac12)\) \(\approx\) \(-0.1497841610 + 0.1894426706i\)
\(L(1)\) \(\approx\) \(0.5656303109 + 0.4858975222i\)
\(L(1)\) \(\approx\) \(0.5656303109 + 0.4858975222i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−30.67323360010025121970799716534, −29.816536343097934737102674146634, −28.65115251216415458853644348227, −27.73197725194663668402820109154, −26.36619068583423175818923001414, −25.10420332429865539829009282019, −23.82433457204451914892591231159, −22.82973311057789702724328009356, −22.036973216067653435167904461507, −20.18786090421116970153479616714, −19.74762170269513199063641939957, −18.328997860946024273857575129675, −18.0671422986092072071400588409, −15.46909240983670763442654450807, −14.60144540690204863010948822612, −13.1424161638793094412941667988, −12.240848657016675659848774461969, −11.37886734169946247758938930956, −9.73920728976908414744945320889, −8.33011797627261958498115398811, −6.922941939468315392551107744038, −5.20721616035079548318015070358, −3.22845079950945900299807899002, −2.27763369503263962051886339492, −0.101819951406014666618136573997, 3.684621539402794409681333527264, 4.29960450749386258287697018902, 5.86132739330498583574200927220, 7.65480645667296317951408253431, 8.587655130836821401198748895162, 9.90685625911992524322252469837, 11.535009202110062955645495252669, 13.193161614366439884827810055340, 14.24800775680518111256706875213, 15.4143359468376678605348151582, 16.55713845103720292025342158679, 16.781563704636980532094121830228, 18.92988955640576369417913187, 20.15762654731370122740687576726, 21.330425177089831222327217716013, 22.43227560802469229593060417883, 23.604916938139320096652964966, 24.3464140096196503505416507869, 25.95131184210594291730719957562, 26.65746822393258506861937789013, 27.37409611473623962935362119019, 28.707513978547129974411424868824, 30.47763465747860996447088243441, 31.710751406270086585153665533773, 32.099317068762231305523970112594

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