Properties

Degree 1
Conductor 71
Sign $0.845 - 0.534i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.753 − 0.657i)2-s + (0.473 + 0.880i)3-s + (0.134 − 0.990i)4-s + (0.309 − 0.951i)5-s + (0.936 + 0.351i)6-s + (−0.393 + 0.919i)7-s + (−0.550 − 0.834i)8-s + (−0.550 + 0.834i)9-s + (−0.393 − 0.919i)10-s + (−0.0448 − 0.998i)11-s + (0.936 − 0.351i)12-s + (−0.0448 + 0.998i)13-s + (0.309 + 0.951i)14-s + (0.983 − 0.178i)15-s + (−0.963 − 0.266i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s,χ)  = 1  + (0.753 − 0.657i)2-s + (0.473 + 0.880i)3-s + (0.134 − 0.990i)4-s + (0.309 − 0.951i)5-s + (0.936 + 0.351i)6-s + (−0.393 + 0.919i)7-s + (−0.550 − 0.834i)8-s + (−0.550 + 0.834i)9-s + (−0.393 − 0.919i)10-s + (−0.0448 − 0.998i)11-s + (0.936 − 0.351i)12-s + (−0.0448 + 0.998i)13-s + (0.309 + 0.951i)14-s + (0.983 − 0.178i)15-s + (−0.963 − 0.266i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.845 - 0.534i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.845 - 0.534i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $0.845 - 0.534i$
motivic weight  =  \(0\)
character  :  $\chi_{71} (38, \cdot )$
Sato-Tate  :  $\mu(35)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 71,\ (0:\ ),\ 0.845 - 0.534i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.422450850 - 0.4117124913i$
$L(\frac12,\chi)$  $\approx$  $1.422450850 - 0.4117124913i$
$L(\chi,1)$  $\approx$  1.512205950 - 0.3332041402i
$L(1,\chi)$  $\approx$  1.512205950 - 0.3332041402i

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

−31.69170164193826853569101994512, −30.64324776817436957713279962887, −30.01012607283993952101330415046, −29.14988184719973245368962852461, −26.9007858788897712287060333895, −25.97107659933591637156975326704, −25.304359672116748344652381087582, −24.147785994473462622548287440059, −22.927784184407709181246463785801, −22.4686466652952705940667614334, −20.658017430275224396307362490807, −19.763166879939363140793314123, −18.01211110672806523271968185728, −17.490685688692484020352002696854, −15.667568998673848427972259769031, −14.59175982200043487406722495772, −13.62097351260479534035808622347, −12.8491110163384312790461984385, −11.32634155875295468883083073614, −9.55233169134642295449894361456, −7.495076765868605999423724060130, −7.18785375545918832911627151252, −5.77361629598589785179374610704, −3.73092196393313027007882215962, −2.478722138741004194894679294603, 2.13376972534334192415471982808, 3.62640442262391043360969168289, 4.94473771624652952970044834809, 6.00253787576065879325161432618, 8.762710444176358579480518660159, 9.40266918251698115807705262049, 10.875766934575846827635494087343, 12.1485201139373237627938318880, 13.38820318329593414389074234473, 14.39914298915943200812610975110, 15.74221136083672873889750051683, 16.51235061424571180576308684084, 18.64017648510016766696808234382, 19.754345924734323676363080977921, 20.7264113820339913329491100674, 21.703942360719768862615663966226, 22.21723423209933619630059602024, 24.0507328675491583896490533016, 24.78448782273356175954536043146, 26.19997628244798317320669197404, 27.58323183034576452473247896578, 28.566620241027623353423842809921, 29.153971903754616763052075546655, 31.052675042737716976617367671329, 31.52652056163862553455855979308

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