Properties

Degree 1
Conductor $ 3 \cdot 7^{2} \cdot 41 $
Sign $0.772 - 0.634i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.237 + 0.971i)2-s + (−0.887 + 0.460i)4-s + (0.119 − 0.992i)5-s + (−0.657 − 0.753i)8-s + (0.992 − 0.119i)10-s + (0.427 − 0.904i)11-s + (0.0224 − 0.999i)13-s + (0.575 − 0.817i)16-s + (0.215 − 0.976i)17-s + (0.998 + 0.0523i)19-s + (0.351 + 0.936i)20-s + (0.979 + 0.200i)22-s + (−0.772 + 0.635i)23-s + (−0.971 − 0.237i)25-s + (0.976 − 0.215i)26-s + ⋯
L(s,χ)  = 1  + (0.237 + 0.971i)2-s + (−0.887 + 0.460i)4-s + (0.119 − 0.992i)5-s + (−0.657 − 0.753i)8-s + (0.992 − 0.119i)10-s + (0.427 − 0.904i)11-s + (0.0224 − 0.999i)13-s + (0.575 − 0.817i)16-s + (0.215 − 0.976i)17-s + (0.998 + 0.0523i)19-s + (0.351 + 0.936i)20-s + (0.979 + 0.200i)22-s + (−0.772 + 0.635i)23-s + (−0.971 − 0.237i)25-s + (0.976 − 0.215i)26-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.772 - 0.634i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.772 - 0.634i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6027\)    =    \(3 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $0.772 - 0.634i$
motivic weight  =  \(0\)
character  :  $\chi_{6027} (11, \cdot )$
Sato-Tate  :  $\mu(840)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6027,\ (0:\ ),\ 0.772 - 0.634i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.826609064 - 0.6536629467i$
$L(\frac12,\chi)$  $\approx$  $1.826609064 - 0.6536629467i$
$L(\chi,1)$  $\approx$  1.176136316 + 0.1199718749i
$L(1,\chi)$  $\approx$  1.176136316 + 0.1199718749i

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

−17.94748983829794360032486264632, −17.4189216129149972524364025993, −16.57332843365367454355264632559, −15.6417828527440514484991465308, −14.807856718013479651057652972100, −14.49183131976206863660203937701, −13.83942109792020640427193272017, −13.215147591242253055156257030097, −12.30569373424287580185271206079, −11.806045266684346091771443317360, −11.3057401816909716630873027466, −10.45386787847786089751500299959, −9.86468388362989443035242018035, −9.562534681069650410485629509890, −8.48736170758275996899483255442, −7.865036742098504722508159397514, −6.69932615948321959168764736187, −6.46526499633828069301366639002, −5.42986654004803764675218906235, −4.62632661350962286872682292786, −3.895287881611223062846096249175, −3.380641069040560204772498964041, −2.34199274046142371330498129887, −1.965544098284483722181870024544, −1.021978533291026184633593410736, 0.58373915012617292570028157145, 1.055900774299823005024423922143, 2.48990263421204929317078038827, 3.46742029265093801302469795337, 3.955473802034298064135738116445, 5.01125022510282663190653925048, 5.431644022812067065740169778101, 5.933800079606689479380921150084, 6.84912986676247604004500169829, 7.709994118901141436057236136310, 8.12541731405399108486618257175, 8.84795801162719204436634766469, 9.517027765817122484699895924033, 9.97740222350928931842320711639, 11.20286823909646991574765116873, 11.911566437812181362019892266835, 12.49545657902653486818316280095, 13.25362500213870708794576566817, 13.77184658456915470780893235508, 14.229354723642588850439119244, 15.17756828185009836965085702705, 15.81427663110345027576903795364, 16.358510924273403917939949633434, 16.69902193054635942493040231159, 17.63502709780778031173590024638

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