Properties

Degree 1
Conductor 67
Sign $0.512 - 0.858i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.0475 − 0.998i)2-s + (0.959 − 0.281i)3-s + (−0.995 + 0.0950i)4-s + (0.654 + 0.755i)5-s + (−0.327 − 0.945i)6-s + (0.888 + 0.458i)7-s + (0.142 + 0.989i)8-s + (0.841 − 0.540i)9-s + (0.723 − 0.690i)10-s + (0.327 − 0.945i)11-s + (−0.928 + 0.371i)12-s + (0.786 + 0.618i)13-s + (0.415 − 0.909i)14-s + (0.841 + 0.540i)15-s + (0.981 − 0.189i)16-s + (−0.995 − 0.0950i)17-s + ⋯
L(s,χ)  = 1  + (−0.0475 − 0.998i)2-s + (0.959 − 0.281i)3-s + (−0.995 + 0.0950i)4-s + (0.654 + 0.755i)5-s + (−0.327 − 0.945i)6-s + (0.888 + 0.458i)7-s + (0.142 + 0.989i)8-s + (0.841 − 0.540i)9-s + (0.723 − 0.690i)10-s + (0.327 − 0.945i)11-s + (−0.928 + 0.371i)12-s + (0.786 + 0.618i)13-s + (0.415 − 0.909i)14-s + (0.841 + 0.540i)15-s + (0.981 − 0.189i)16-s + (−0.995 − 0.0950i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.512 - 0.858i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.512 - 0.858i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(67\)
\( \varepsilon \)  =  $0.512 - 0.858i$
motivic weight  =  \(0\)
character  :  $\chi_{67} (57, \cdot )$
Sato-Tate  :  $\mu(66)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 67,\ (1:\ ),\ 0.512 - 0.858i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.053124559 - 1.165340082i$
$L(\frac12,\chi)$  $\approx$  $2.053124559 - 1.165340082i$
$L(\chi,1)$  $\approx$  1.447332850 - 0.6402355494i
$L(1,\chi)$  $\approx$  1.447332850 - 0.6402355494i

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

−32.16661810933429359038185290180, −31.09266039617079411081446274712, −30.133411213790056797458417087927, −28.11389014705343013054069036850, −27.43431085053744333269891506681, −26.13227951314061829268487722234, −25.30697506871865926681447292914, −24.51841086155975017073631766271, −23.42105110964456150281696148030, −21.82090645006193015974141497298, −20.760897102397625659297858909574, −19.72738580028468164126078356377, −17.96266260051461511144836867768, −17.246320621023520614692801125707, −15.793418548194220433162586720377, −14.83511773405893844068460967016, −13.71626315633772617683494562605, −12.90995984374980327515763491096, −10.4323704053931455623911388761, −9.10887496273337507110254029683, −8.33699178716704249126207319322, −6.98523677671516443108619734513, −5.12977706022482176168008186185, −4.11040256699318096082758318067, −1.595376956947970771333327658835, 1.64432560349006195950478585190, 2.70485259546721020940846672255, 4.19895119848467529229796947259, 6.28113920076748407930594759408, 8.28526877678757189935517428987, 9.08885718757774996610017684204, 10.57399072050749982420423287320, 11.6559388935401997690691950904, 13.318376311644990642094911428749, 14.04061547942153318203246418150, 15.04487129918971203182598474799, 17.30189488508258339151591837157, 18.63650821516205403249451384257, 18.904994989305077371161113257450, 20.57559424463129473458676022634, 21.27871286301333277091811704363, 22.22058998678293609658910765156, 23.86604738815398460189540617991, 25.06816372371282228044743714086, 26.3659899533914521012714383338, 27.02046845912931125605377108328, 28.47097860438442492558954562118, 29.69516305115342956082107169798, 30.43122133162732029637395346021, 31.21213069784480671633904143837

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