Properties

Degree 1
Conductor 71
Sign $-0.994 - 0.104i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)3-s + (−0.900 + 0.433i)4-s + 5-s + (0.623 + 0.781i)6-s + (0.222 − 0.974i)7-s + (0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (−0.222 − 0.974i)10-s + (−0.623 + 0.781i)11-s + (0.623 − 0.781i)12-s + (−0.623 − 0.781i)13-s − 14-s + (−0.900 + 0.433i)15-s + (0.623 − 0.781i)16-s − 17-s + ⋯
L(s,χ)  = 1  + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)3-s + (−0.900 + 0.433i)4-s + 5-s + (0.623 + 0.781i)6-s + (0.222 − 0.974i)7-s + (0.623 + 0.781i)8-s + (0.623 − 0.781i)9-s + (−0.222 − 0.974i)10-s + (−0.623 + 0.781i)11-s + (0.623 − 0.781i)12-s + (−0.623 − 0.781i)13-s − 14-s + (−0.900 + 0.433i)15-s + (0.623 − 0.781i)16-s − 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.994 - 0.104i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.994 - 0.104i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $-0.994 - 0.104i$
motivic weight  =  \(0\)
character  :  $\chi_{71} (34, \cdot )$
Sato-Tate  :  $\mu(14)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 71,\ (1:\ ),\ -0.994 - 0.104i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.03026940503 - 0.5800576189i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.03026940503 - 0.5800576189i\)
\(L(\chi,1)\)  \(\approx\)  \(0.5451470416 - 0.3485967357i\)
\(L(1,\chi)\)  \(\approx\)  \(0.5451470416 - 0.3485967357i\)

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

−32.12832174295611522274227484129, −31.09775778355799961661848165303, −29.40089468531407707396984392650, −28.710908358878575636033110190101, −27.65563028867350616050754803433, −26.38904066634570695166023446367, −25.13086641920926472353539331854, −24.42116390536166588247162558525, −23.497751698047127673426875884972, −22.00348090746053651149503202209, −21.5711600436127037505892611254, −19.05151476395484474400459621440, −18.305049781989615230020916201457, −17.3475196939135442835263780814, −16.45829827289315699819644884034, −15.16055604689849777525751662098, −13.74020313820656603388816409290, −12.77881949868865648291258081485, −11.12633744849656640088049331767, −9.69144717858193240705632621974, −8.42268642609470642215063442504, −6.80174352191739566565329752587, −5.824312048091223050402158142764, −4.94850735082202834557817611553, −1.87076130334998205898871250506, 0.34218186441511591811071204398, 2.18154797269166736545828905853, 4.24290746159099869342305875912, 5.302151904023522540632111445763, 7.15463682853156870134536431582, 9.1542419408343556827685679138, 10.42985280639172658365549185571, 10.73939094609611475330256981298, 12.51532611317130107167094353077, 13.32035142006070395036571331768, 14.92566729974003716231439423072, 16.87294406575155776164289447031, 17.47696957805970232642262451614, 18.33253850002787630603217387526, 20.17486919545426174940865150897, 20.87582747596529140640304943715, 22.04540962000321725255204913311, 22.78058279548136182645897077616, 24.03985252127337384988667256873, 25.89401249365420108765851976218, 26.80360898827848279988512518764, 27.89588420054443048832547929675, 28.84732131899949567816158378698, 29.60080064882580116991889785882, 30.446869781449706259440408267848

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