Properties

Degree 1
Conductor $ 37 \cdot 109 $
Sign $-0.972 - 0.230i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.642 − 0.766i)2-s + (−0.686 + 0.727i)3-s + (−0.173 − 0.984i)4-s + (0.893 − 0.448i)5-s + (0.116 + 0.993i)6-s + (−0.835 − 0.549i)7-s + (−0.866 − 0.5i)8-s + (−0.0581 − 0.998i)9-s + (0.230 − 0.973i)10-s + (0.230 + 0.973i)11-s + (0.835 + 0.549i)12-s + (0.549 − 0.835i)13-s + (−0.957 + 0.286i)14-s + (−0.286 + 0.957i)15-s + (−0.939 + 0.342i)16-s + (0.342 + 0.939i)17-s + ⋯
L(s,χ)  = 1  + (0.642 − 0.766i)2-s + (−0.686 + 0.727i)3-s + (−0.173 − 0.984i)4-s + (0.893 − 0.448i)5-s + (0.116 + 0.993i)6-s + (−0.835 − 0.549i)7-s + (−0.866 − 0.5i)8-s + (−0.0581 − 0.998i)9-s + (0.230 − 0.973i)10-s + (0.230 + 0.973i)11-s + (0.835 + 0.549i)12-s + (0.549 − 0.835i)13-s + (−0.957 + 0.286i)14-s + (−0.286 + 0.957i)15-s + (−0.939 + 0.342i)16-s + (0.342 + 0.939i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.972 - 0.230i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.972 - 0.230i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $-0.972 - 0.230i$
motivic weight  =  \(0\)
character  :  $\chi_{4033} (10, \cdot )$
Sato-Tate  :  $\mu(108)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4033,\ (1:\ ),\ -0.972 - 0.230i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2786422061 - 2.380180210i$
$L(\frac12,\chi)$  $\approx$  $0.2786422061 - 2.380180210i$
$L(\chi,1)$  $\approx$  1.058283752 - 0.6746075726i
$L(1,\chi)$  $\approx$  1.058283752 - 0.6746075726i

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

−18.45822079550934013226654268385, −18.00225390613701829042721960407, −17.05097123915701343213674857805, −16.46634694556244936006577756746, −16.197112630724447657513707417, −15.24304597925440463409029312527, −14.18706192712656218807690508928, −13.86284225818599863347236458361, −13.336334656545397859383091937621, −12.56996616540340675789206113940, −11.89569767523474752930707157101, −11.34783893190411054057843610, −10.42080359097025871387411079528, −9.42681370760969273371184511086, −8.85996668903944534749343712790, −7.89100892887206565944563665383, −7.152504079669119506294466657995, −6.37521803462147068946454081675, −5.97420163383377147470297623064, −5.66159373576996125050992648767, −4.60800462401404406624548747475, −3.54455561416898348060515873675, −2.8178944236901753297006433936, −2.02153645672491911594975328216, −0.91647986634333313520902273890, 0.385426139504014081529958160999, 0.958408013159853079069423322465, 2.01667777003701784911721731948, 2.91558415098678058907144798034, 3.81070202641988386804249892441, 4.39702159500262969912002365442, 5.0528488985943640857071410259, 5.990809681184552608546131088888, 6.223936205750528012395942178818, 7.12968082869581926072785857593, 8.68127257484823631074110325241, 9.20721575181758450500299294527, 10.15492999873398644271802154241, 10.28981113565423843600606273009, 10.831250917856308722462325973180, 12.05865222985709179882745947966, 12.47709488187284586178839671638, 13.00723540937684023254989594456, 13.779674338359385041904993056575, 14.4605582278750683636691940777, 15.38043893820607750802468014971, 15.75685726418510914107212280018, 16.66569533638809691487535484556, 17.49950411040539137574970554610, 17.69541884325799700116959993

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