Properties

Degree 1
Conductor 67
Sign $0.611 + 0.791i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.415 + 0.909i)2-s + (−0.841 + 0.540i)3-s + (−0.654 − 0.755i)4-s + (0.142 − 0.989i)5-s + (−0.142 − 0.989i)6-s + (−0.415 + 0.909i)7-s + (0.959 − 0.281i)8-s + (0.415 − 0.909i)9-s + (0.841 + 0.540i)10-s + (0.142 − 0.989i)11-s + (0.959 + 0.281i)12-s + (0.959 + 0.281i)13-s + (−0.654 − 0.755i)14-s + (0.415 + 0.909i)15-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)17-s + ⋯
L(s,χ)  = 1  + (−0.415 + 0.909i)2-s + (−0.841 + 0.540i)3-s + (−0.654 − 0.755i)4-s + (0.142 − 0.989i)5-s + (−0.142 − 0.989i)6-s + (−0.415 + 0.909i)7-s + (0.959 − 0.281i)8-s + (0.415 − 0.909i)9-s + (0.841 + 0.540i)10-s + (0.142 − 0.989i)11-s + (0.959 + 0.281i)12-s + (0.959 + 0.281i)13-s + (−0.654 − 0.755i)14-s + (0.415 + 0.909i)15-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.611 + 0.791i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.611 + 0.791i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(67\)
\( \varepsilon \)  =  $0.611 + 0.791i$
motivic weight  =  \(0\)
character  :  $\chi_{67} (52, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 67,\ (1:\ ),\ 0.611 + 0.791i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.8116531719 + 0.3987034370i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.8116531719 + 0.3987034370i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6572560285 + 0.2728139545i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6572560285 + 0.2728139545i\)

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

−30.838377722334014048467473476420, −30.368318628639455109088907723193, −29.38000213596219113568092225711, −28.57679735542988896433855206878, −27.376521137303181479975606207191, −26.32491179549182153724108969248, −25.23596558462286722889549745578, −23.12060781140854339647929272744, −22.89121532897966774566252194353, −21.66563162388794442272793741287, −20.16972878173570529543340706435, −19.1507940080459268388326788652, −17.92811992338104209192334819604, −17.48915089173315130065818512886, −15.91129148849787355940612476921, −13.81186326979104223938697532849, −12.99283871101191636647507998859, −11.4915260528231276529844982491, −10.745731703396220484131842802690, −9.64597465150373120816160606893, −7.558738651970168166263226805874, −6.63046748934134493391855797033, −4.521637013776537375455120645831, −2.77913276182525733877962302614, −0.96986378232948254441805470593, 0.89387759782224770075762029247, 4.205496132528574809974378771732, 5.64869568862312627907248781825, 6.26266012501577742139049237914, 8.4986831516969910905864156956, 9.22137294126389853142127987457, 10.6663502527538072276007592324, 12.20721726003103513663913944886, 13.5651034211689512681095056738, 15.307660529267382847648551351368, 16.16411740812216672996840084661, 16.85590071526772690580217720374, 18.0910251742620347086190020505, 19.20040334446565784444926953383, 20.972298199913033136695148691532, 22.051082489828959669311140299565, 23.235577867630861562697473078693, 24.26925701133763287875303525405, 25.16372840662106425925318692467, 26.52897632737566445349925825246, 27.554106721410904530253531466893, 28.54322543367339695371086253711, 28.9798826890412348441210562655, 31.27818735316315867352850659827, 32.40175924599560664561268468714

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