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Properties

Label 1-71-71.42-r1-0-0
Degree $1$
Conductor $71$
Sign $-0.498 + 0.866i$
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  + (0.473 − 0.880i)2-s + (−0.0448 + 0.998i)3-s + (−0.550 − 0.834i)4-s + (0.309 + 0.951i)5-s + (0.858 + 0.512i)6-s + (−0.983 + 0.178i)7-s + (−0.995 + 0.0896i)8-s + (−0.995 − 0.0896i)9-s + (0.983 + 0.178i)10-s + (−0.753 − 0.657i)11-s + (0.858 − 0.512i)12-s + (−0.753 + 0.657i)13-s + (−0.309 + 0.951i)14-s + (−0.963 + 0.266i)15-s + (−0.393 + 0.919i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (0.473 − 0.880i)2-s + (−0.0448 + 0.998i)3-s + (−0.550 − 0.834i)4-s + (0.309 + 0.951i)5-s + (0.858 + 0.512i)6-s + (−0.983 + 0.178i)7-s + (−0.995 + 0.0896i)8-s + (−0.995 − 0.0896i)9-s + (0.983 + 0.178i)10-s + (−0.753 − 0.657i)11-s + (0.858 − 0.512i)12-s + (−0.753 + 0.657i)13-s + (−0.309 + 0.951i)14-s + (−0.963 + 0.266i)15-s + (−0.393 + 0.919i)16-s + (0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $-0.498 + 0.866i$
Analytic conductor: \(7.63000\)
Root analytic conductor: \(7.63000\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (1:\ ),\ -0.498 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3907781487 + 0.6754154948i\)
\(L(\frac12)\) \(\approx\) \(0.3907781487 + 0.6754154948i\)
\(L(1)\) \(\approx\) \(0.8853921287 + 0.1108345086i\)
\(L(1)\) \(\approx\) \(0.8853921287 + 0.1108345086i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.473 - 0.880i)T \)
3 \( 1 + (-0.0448 + 0.998i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
7 \( 1 + (-0.983 + 0.178i)T \)
11 \( 1 + (-0.753 - 0.657i)T \)
13 \( 1 + (-0.753 + 0.657i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.963 - 0.266i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
29 \( 1 + (0.936 + 0.351i)T \)
31 \( 1 + (0.393 + 0.919i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (-0.691 - 0.722i)T \)
47 \( 1 + (0.0448 + 0.998i)T \)
53 \( 1 + (0.550 - 0.834i)T \)
59 \( 1 + (-0.858 + 0.512i)T \)
61 \( 1 + (-0.983 - 0.178i)T \)
67 \( 1 + (0.550 + 0.834i)T \)
73 \( 1 + (0.473 - 0.880i)T \)
79 \( 1 + (-0.995 + 0.0896i)T \)
83 \( 1 + (0.858 - 0.512i)T \)
89 \( 1 + (-0.550 + 0.834i)T \)
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

−31.49045544632563583668454502055, −29.98324627881309999812098099507, −29.15236757222874011696957465682, −27.89264273255024648739806515131, −26.23410262116037309754863179808, −25.203813007327443586976771735414, −24.72050079466150648026578527569, −23.322528726956587698696477675005, −22.878191852745495943953282802287, −21.21050348201800019032409839998, −19.95363604083545468734905634295, −18.55793268125365599052115977377, −17.30321015972925553290109694895, −16.601897591295355558005021192055, −15.20528749953342781994419765926, −13.72923352693698148431232683734, −12.78468737002705851467739472247, −12.32113374706945915390697401397, −9.79870955828337879183327730811, −8.35328294812501694570829539193, −7.27601288636207700441917901999, −6.02374032607937946239585210728, −4.8539799570435170165434617412, −2.767540905710711288152705789341, −0.31875466150606982453300613432, 2.65801386164465939372827684610, 3.53426701697204421567184642164, 5.19414494264419720701697672938, 6.43701364718631279273678909623, 8.94436238136587559383488196904, 10.14956984576287604154405832018, 10.72335019061080227078299593450, 12.15234219921439427333459593592, 13.62995891524664019757998000382, 14.68461762986666157995749595534, 15.704282972202252679374014802766, 17.24150198014658587836364717403, 18.897886424687589607938734174544, 19.503979007162919071294994599277, 21.26241925745657838170031301557, 21.62180299067541676718330097317, 22.68668444139539067752017495040, 23.56978783725720757643757684280, 25.60919106056599228345248648375, 26.53296613551612629404713449437, 27.50455991393795372476258861513, 28.90912341346249629922402608592, 29.33833262358785755757695026661, 30.79826678525312277864247616612, 31.88478472991462483266302700064

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