Properties

Degree 1
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $-0.188 - 0.982i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.280 − 0.959i)2-s + (−0.842 − 0.538i)4-s + (−0.599 + 0.800i)5-s + (−0.753 + 0.657i)8-s + (0.599 + 0.800i)10-s + (−0.646 + 0.762i)11-s + (−0.691 − 0.722i)13-s + (0.420 + 0.907i)16-s + (−0.887 − 0.460i)17-s + (−0.978 + 0.207i)19-s + (0.936 − 0.351i)20-s + (0.550 + 0.834i)22-s + (0.163 + 0.986i)23-s + (−0.280 − 0.959i)25-s + (−0.887 + 0.460i)26-s + ⋯
L(s,χ)  = 1  + (0.280 − 0.959i)2-s + (−0.842 − 0.538i)4-s + (−0.599 + 0.800i)5-s + (−0.753 + 0.657i)8-s + (0.599 + 0.800i)10-s + (−0.646 + 0.762i)11-s + (−0.691 − 0.722i)13-s + (0.420 + 0.907i)16-s + (−0.887 − 0.460i)17-s + (−0.978 + 0.207i)19-s + (0.936 − 0.351i)20-s + (0.550 + 0.834i)22-s + (0.163 + 0.986i)23-s + (−0.280 − 0.959i)25-s + (−0.887 + 0.460i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.188 - 0.982i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.188 - 0.982i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $-0.188 - 0.982i$
motivic weight  =  \(0\)
character  :  $\chi_{6027} (1958, \cdot )$
Sato-Tate  :  $\mu(210)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6027,\ (0:\ ),\ -0.188 - 0.982i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2692789284 - 0.3258780523i$
$L(\frac12,\chi)$  $\approx$  $0.2692789284 - 0.3258780523i$
$L(\chi,1)$  $\approx$  0.6544876599 - 0.2125583202i
$L(1,\chi)$  $\approx$  0.6544876599 - 0.2125583202i

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

−17.61335775639787102850625270550, −17.04908697368334815800979665501, −16.46129348075785221990892013974, −16.04093957366928250617241982149, −15.32518983019944185357812025840, −14.70748132607103385773828776227, −14.11760163746335869452243033211, −13.17658124119901943407023169787, −12.782364846183678071637101896627, −12.332616508106269315397717745231, −11.21117483840563586811811369057, −10.836371292318556378032554420667, −9.53518535000237815326696365023, −9.02401381350577178349267828214, −8.489964733960867224465575384260, −7.78919820580303244241091579830, −7.224968854505346364690147444747, −6.35286223241574657316824074870, −5.74963659909467947460231412089, −4.89602393880207159574744341500, −4.37471498928255010454797560218, −3.823555043488960376269090610825, −2.79917690693418279709457052105, −1.8502022431379159782702210408, −0.45289291839381709088731510188, 0.19357561865155848509766777714, 1.58017363229472455779081917443, 2.447171203879749619101696647452, 2.80849498016249545639492197292, 3.80529758816692430920046855300, 4.2988557391339140610356717815, 5.176706528648222379736197582850, 5.76067526920275217709206402645, 6.876115671748311558777302619152, 7.408753946322084479969209287449, 8.20044580808170595943022475592, 8.99162143532426281021983743837, 9.85332272650567392448865113195, 10.31825260250685762132358469225, 10.96273584070447799100008862670, 11.53918674296759969164240545992, 12.189966673843873887522501022576, 12.962421559481393704102698650655, 13.30688490500313428782059189628, 14.299441287064166011907162442892, 14.95825507008393474775034078157, 15.23538477738513275173901487833, 15.99424544316627436965197233617, 17.213887712845389734458382637723, 17.671148296480557184477061830438

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