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Properties

Label 1-71-71.56-r1-0-0
Degree $1$
Conductor $71$
Sign $0.0279 + 0.999i$
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  + (−0.691 − 0.722i)2-s + (0.936 + 0.351i)3-s + (−0.0448 + 0.998i)4-s + (−0.809 − 0.587i)5-s + (−0.393 − 0.919i)6-s + (−0.134 + 0.990i)7-s + (0.753 − 0.657i)8-s + (0.753 + 0.657i)9-s + (0.134 + 0.990i)10-s + (−0.858 + 0.512i)11-s + (−0.393 + 0.919i)12-s + (−0.858 − 0.512i)13-s + (0.809 − 0.587i)14-s + (−0.550 − 0.834i)15-s + (−0.995 − 0.0896i)16-s + (−0.309 + 0.951i)17-s + ⋯
L(s)  = 1  + (−0.691 − 0.722i)2-s + (0.936 + 0.351i)3-s + (−0.0448 + 0.998i)4-s + (−0.809 − 0.587i)5-s + (−0.393 − 0.919i)6-s + (−0.134 + 0.990i)7-s + (0.753 − 0.657i)8-s + (0.753 + 0.657i)9-s + (0.134 + 0.990i)10-s + (−0.858 + 0.512i)11-s + (−0.393 + 0.919i)12-s + (−0.858 − 0.512i)13-s + (0.809 − 0.587i)14-s + (−0.550 − 0.834i)15-s + (−0.995 − 0.0896i)16-s + (−0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0279 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0279 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.0279 + 0.999i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ 0.0279 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5447315995 + 0.5601778326i\)
\(L(\frac12)\) \(\approx\) \(0.5447315995 + 0.5601778326i\)
\(L(1)\) \(\approx\) \(0.7481058003 + 0.06416284189i\)
\(L(1)\) \(\approx\) \(0.7481058003 + 0.06416284189i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.691 - 0.722i)T \)
3 \( 1 + (0.936 + 0.351i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.134 + 0.990i)T \)
11 \( 1 + (-0.858 + 0.512i)T \)
13 \( 1 + (-0.858 - 0.512i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.550 + 0.834i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
29 \( 1 + (-0.963 + 0.266i)T \)
31 \( 1 + (0.995 - 0.0896i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (0.983 + 0.178i)T \)
47 \( 1 + (-0.936 + 0.351i)T \)
53 \( 1 + (0.0448 + 0.998i)T \)
59 \( 1 + (0.393 - 0.919i)T \)
61 \( 1 + (-0.134 - 0.990i)T \)
67 \( 1 + (0.0448 - 0.998i)T \)
73 \( 1 + (-0.691 - 0.722i)T \)
79 \( 1 + (0.753 - 0.657i)T \)
83 \( 1 + (-0.393 + 0.919i)T \)
89 \( 1 + (-0.0448 - 0.998i)T \)
97 \( 1 + (0.222 + 0.974i)T \)
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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

−31.41710184619273682560797247809, −30.0342249238591229440210787304, −29.06658590635798068899023605792, −27.33827358963012531297270697285, −26.47598525313702653203083152072, −26.16477175358370971814862438403, −24.57004503145385628937058304696, −23.83034459886451121257062336580, −22.79718272426668044668576188130, −20.78718804835901568941903805826, −19.55777602445639608902016493498, −19.05848069100349203045555221526, −17.85039203047394144008512983071, −16.38000220670173353649343413527, −15.30351069610158334913188863741, −14.32668594569577505215143629234, −13.2697626346766305411380966387, −11.208070265253439583314158152371, −9.97919210083853916376281982934, −8.606255786954343378147270845642, −7.413750059242400085386437119045, −6.87686647410489039146934379081, −4.50590870323437545194926130664, −2.68630463040575195042504054764, −0.42613568408587043024800450937, 2.04206599989564652426899806946, 3.361088064345328452403322965742, 4.814303294766704410172769844888, 7.60309254604391605517398309278, 8.41961468666140089606580942525, 9.4509163538910224670398005837, 10.681797676650699863041463133874, 12.33751950090778536046691264838, 12.95156708932447208563953022709, 15.04541072849591215665954442045, 15.80036467401144282800367611569, 17.22611943805769314623802799678, 18.83080325429181185504202046128, 19.421358794636754816023014529, 20.55613494882580159668408771700, 21.30020844753810277740662025009, 22.58654427767292143146039256854, 24.4100818243288893550353207127, 25.429144855619855321445465642465, 26.42659941688220895013101299727, 27.540600892979506408869626899388, 28.14420413849524611304012551391, 29.434906344150716490682173149097, 30.997636759658696281467391203252, 31.334486826256180777906215789707

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