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Properties

Label 1-71-71.67-r1-0-0
Degree $1$
Conductor $71$
Sign $0.811 - 0.583i$
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  + (0.983 + 0.178i)2-s + (−0.963 + 0.266i)3-s + (0.936 + 0.351i)4-s + (0.309 − 0.951i)5-s + (−0.995 + 0.0896i)6-s + (−0.473 − 0.880i)7-s + (0.858 + 0.512i)8-s + (0.858 − 0.512i)9-s + (0.473 − 0.880i)10-s + (0.393 − 0.919i)11-s + (−0.995 − 0.0896i)12-s + (0.393 + 0.919i)13-s + (−0.309 − 0.951i)14-s + (−0.0448 + 0.998i)15-s + (0.753 + 0.657i)16-s + (0.809 − 0.587i)17-s + ⋯
L(s)  = 1  + (0.983 + 0.178i)2-s + (−0.963 + 0.266i)3-s + (0.936 + 0.351i)4-s + (0.309 − 0.951i)5-s + (−0.995 + 0.0896i)6-s + (−0.473 − 0.880i)7-s + (0.858 + 0.512i)8-s + (0.858 − 0.512i)9-s + (0.473 − 0.880i)10-s + (0.393 − 0.919i)11-s + (−0.995 − 0.0896i)12-s + (0.393 + 0.919i)13-s + (−0.309 − 0.951i)14-s + (−0.0448 + 0.998i)15-s + (0.753 + 0.657i)16-s + (0.809 − 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $0.811 - 0.583i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ 0.811 - 0.583i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.201544748 - 0.7093912126i\)
\(L(\frac12)\) \(\approx\) \(2.201544748 - 0.7093912126i\)
\(L(1)\) \(\approx\) \(1.575691748 - 0.1824923047i\)
\(L(1)\) \(\approx\) \(1.575691748 - 0.1824923047i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.983 + 0.178i)T \)
3 \( 1 + (-0.963 + 0.266i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (-0.473 - 0.880i)T \)
11 \( 1 + (0.393 - 0.919i)T \)
13 \( 1 + (0.393 + 0.919i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.0448 - 0.998i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
29 \( 1 + (-0.550 - 0.834i)T \)
31 \( 1 + (-0.753 + 0.657i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (0.134 + 0.990i)T \)
47 \( 1 + (0.963 + 0.266i)T \)
53 \( 1 + (-0.936 + 0.351i)T \)
59 \( 1 + (0.995 + 0.0896i)T \)
61 \( 1 + (-0.473 + 0.880i)T \)
67 \( 1 + (-0.936 - 0.351i)T \)
73 \( 1 + (0.983 + 0.178i)T \)
79 \( 1 + (0.858 + 0.512i)T \)
83 \( 1 + (-0.995 - 0.0896i)T \)
89 \( 1 + (0.936 - 0.351i)T \)
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

−31.36204357596145764323162364481, −30.30419502619791140978282238024, −29.62057303995202433958770376742, −28.55347797062718379444467472415, −27.616213618283718219843780729717, −25.57153588542734292059530402007, −24.98710924417779366231140571729, −23.42230828899378980509756970301, −22.61382513768809092491771029268, −22.08284618747630762989485836682, −20.84311834268289213583403304155, −19.1537487568447110486972188799, −18.270479761940994354021562955361, −16.83434696194042310368189702963, −15.45309464203994398740527307125, −14.58565932911506439460302628376, −12.91854032686866053604362864125, −12.23917051496583162777410954865, −10.92713764408183662304224967613, −9.98751405815799636516972454973, −7.34492727407821989658317817724, −6.20219199770779884591142578483, −5.39987184028395301146978586003, −3.497413008505889064944366732927, −1.87218477724955325206865122422, 1.069311816838886668374499597319, 3.689651854437828427604741320251, 4.854886860479310613886476916083, 6.0037762423858644434793680567, 7.16641595605316905275879779659, 9.25865814676062859161796256111, 10.859951585871624007482335679467, 11.86479538202910738769242636415, 13.091392504509484978181184419425, 13.952766695057222930944894806723, 15.816105589983013694126857729916, 16.57884509612366381402956577446, 17.228421109420185333790026250386, 19.29080877263544118686861624015, 20.74040062085259732376456722778, 21.46323289768027216598477586148, 22.61534222263487729866071320426, 23.66912800169882828668692684603, 24.25831895323378110345504898261, 25.67034937849666540965326348452, 27.02128256396921381353676589811, 28.45038021337380711263565570774, 29.283114583371187503667265697144, 30.008872202673614255985639639085, 31.64808485484989310612455767358

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