Properties

Degree 1
Conductor 71
Sign $-0.498 + 0.866i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $-0.498 + 0.866i$
motivic weight  =  \(0\)
character  :  $\chi_{71} (42, \cdot )$
Sato-Tate  :  $\mu(70)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 71,\ (1:\ ),\ -0.498 + 0.866i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.3907781487 + 0.6754154948i$
$L(\frac12,\chi)$  $\approx$  $0.3907781487 + 0.6754154948i$
$L(\chi,1)$  $\approx$  0.8853921287 + 0.1108345086i
$L(1,\chi)$  $\approx$  0.8853921287 + 0.1108345086i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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