Properties

Degree 1
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $-0.317 + 0.948i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.925 − 0.379i)2-s + (0.712 − 0.701i)4-s + (0.193 − 0.981i)5-s + (0.393 − 0.919i)8-s + (−0.193 − 0.981i)10-s + (−0.999 − 0.0299i)11-s + (0.134 − 0.990i)13-s + (0.0149 − 0.999i)16-s + (−0.251 + 0.967i)17-s + (−0.978 + 0.207i)19-s + (−0.550 − 0.834i)20-s + (−0.936 + 0.351i)22-s + (−0.998 − 0.0598i)23-s + (−0.925 − 0.379i)25-s + (−0.251 − 0.967i)26-s + ⋯
L(s,χ)  = 1  + (0.925 − 0.379i)2-s + (0.712 − 0.701i)4-s + (0.193 − 0.981i)5-s + (0.393 − 0.919i)8-s + (−0.193 − 0.981i)10-s + (−0.999 − 0.0299i)11-s + (0.134 − 0.990i)13-s + (0.0149 − 0.999i)16-s + (−0.251 + 0.967i)17-s + (−0.978 + 0.207i)19-s + (−0.550 − 0.834i)20-s + (−0.936 + 0.351i)22-s + (−0.998 − 0.0598i)23-s + (−0.925 − 0.379i)25-s + (−0.251 − 0.967i)26-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.317 + 0.948i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.317 + 0.948i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $-0.317 + 0.948i$
motivic weight  =  \(0\)
character  :  $\chi_{6027} (236, \cdot )$
Sato-Tate  :  $\mu(210)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6027,\ (0:\ ),\ -0.317 + 0.948i)$
$L(\chi,\frac{1}{2})$  $\approx$  $-0.3146275008 - 0.4372288676i$
$L(\frac12,\chi)$  $\approx$  $-0.3146275008 - 0.4372288676i$
$L(\chi,1)$  $\approx$  1.202170617 - 0.7179379641i
$L(1,\chi)$  $\approx$  1.202170617 - 0.7179379641i

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

−18.21361773038407652437016783928, −17.44585931455533781139770226129, −16.65528185298049816289769279223, −16.103471088218863403523186373606, −15.35230493291603001932380530625, −14.935440419510658861221742710910, −14.21216691473220487460517319449, −13.50105591254801251247770135771, −13.316738019234037803322479643658, −12.233058947069677418042180295269, −11.60644473958625082202554296794, −11.07199753991614964594864376599, −10.38997090165154179750378886215, −9.631579084335507728815250381376, −8.68564610844075238265934263976, −7.8318758203145268731745609223, −7.28644408280044587508337252966, −6.64545610558824954864488089538, −5.992058692644901342328060957791, −5.40442175421023556154881221300, −4.34817495965282610770061937234, −4.02575097659561647073279404902, −2.886843881861391227264717823466, −2.456176510562811009762622928662, −1.79495350757003456844103135152, 0.082247281089419295002492343378, 1.19431133616464511751575943349, 1.998526998231042876922147708366, 2.60452084680828569348424526411, 3.72464342772009978121195012327, 4.093575972643477958691613329809, 5.19628266984826630397666695663, 5.39916892628053634796359246441, 6.15577319914489284833431949714, 6.97206830264367422230479494258, 8.0690334488419286699333549003, 8.33932202618788209965334576920, 9.40406235723177357011180415908, 10.2269083292095186718253628356, 10.61439327659426168426599028204, 11.33555707784923195812905315405, 12.35064176548472039347138066928, 12.73814684958573325196042918135, 13.07644271309785745267116137265, 13.79404232522092466672407385100, 14.66487569159070269787062524120, 15.17659338803282252924206655606, 15.93159731125919844375920810013, 16.35915099702411832483811332544, 17.15365680886407323877709150031

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