Properties

Degree 1
Conductor $ 3 \cdot 7 $
Sign $0.605 - 0.795i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 8-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s − 13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + 20-s + 22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s,χ)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s − 8-s + (0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s − 13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + 20-s + 22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.605 - 0.795i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 21 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.605 - 0.795i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(21\)    =    \(3 \cdot 7\)
\( \varepsilon \)  =  $0.605 - 0.795i$
motivic weight  =  \(0\)
character  :  $\chi_{21} (17, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 21,\ (0:\ ),\ 0.605 - 0.795i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6694938424 - 0.3318863448i$
$L(\frac12,\chi)$  $\approx$  $0.6694938424 - 0.3318863448i$
$L(\chi,1)$  $\approx$  0.9424309741 - 0.3602888690i
$L(1,\chi)$  $\approx$  0.9424309741 - 0.3602888690i

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

−39.9757841221087542132090392787, −39.245778385534817742539824247568, −37.12049190455360556345390151354, −35.64257411497973349963621131466, −34.83684405639583132685959043978, −33.28830860361225655819214834053, −32.093169105228246916449007509745, −31.23783466594073402673008418062, −29.54949492304203645567746298403, −27.66619023623240865386270687932, −26.5239049066800223364054045728, −24.76209333329044632231109900533, −24.05665419571425407988859176941, −22.53845428531322685194775289527, −21.13548794719424623266817352206, −19.378901146721629871300690671900, −17.2924857541050276153024313542, −16.25833267856615512172453481886, −14.8119131191553031284187735102, −13.19241251818307983248006908592, −11.8376479720516422789104848514, −9.05833888683324419147075977365, −7.63347520175472358408746225190, −5.63227241518595499288917969016, −3.93679131959732380659178331816, 2.713661054530821908489531505200, 4.61676697612220809176794508841, 6.963051664297083192261294811440, 9.535060568676000726486421312049, 11.07663219778130032536446790237, 12.340493546549576107432982097443, 14.16776444266309741666574815350, 15.31430467617490589662058340581, 17.8099935932126404559635957513, 19.21531180032017935354707530881, 20.33203324297432567727436708573, 22.1242055806246711510240906359, 22.86190794944307696502341998623, 24.46857275710259925298939854837, 26.538738063046226704313142259794, 27.72055470111870522019971924738, 29.21052816771762619536447304847, 30.44017502751045189140085211736, 31.32619916618084365232408180972, 32.833383076707142705541313466708, 34.21435508975330046908853886670, 35.89853130034376326083308420766, 37.35275904908157588671755386615, 38.48269784891810374623718432223, 39.30510211146014997278007439712

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