Properties

Degree 1
Conductor 1009
Sign $-0.245 + 0.969i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 + 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + 13-s + 14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯
L(s,χ)  = 1  + (−0.5 + 0.866i)2-s + 3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + 9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)12-s + 13-s + 14-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.245 + 0.969i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1009 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.245 + 0.969i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1009\)
\( \varepsilon \)  =  $-0.245 + 0.969i$
motivic weight  =  \(0\)
character  :  $\chi_{1009} (374, \cdot )$
Sato-Tate  :  $\mu(3)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1009,\ (0:\ ),\ -0.245 + 0.969i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8398666410 + 1.078843466i$
$L(\frac12,\chi)$  $\approx$  $0.8398666410 + 1.078843466i$
$L(\chi,1)$  $\approx$  0.8962950056 + 0.5151133994i
$L(1,\chi)$  $\approx$  0.8962950056 + 0.5151133994i

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

−20.908547909330464969983663201225, −20.75089814618216075528001489925, −19.844751640162877142367991390158, −19.07620377036106747168709970548, −18.65179084545895075442380239183, −17.8621035029502554715509258048, −16.48467752077884604276502856029, −15.97447169826756324468078606976, −15.36657129392353156289528837563, −13.90175597478825371855176539201, −13.29412413326781679402374624823, −12.75279193031547909636197508916, −11.76580794668900549432118046498, −11.10905890889562539272559949125, −9.88743720823488643851382125911, −9.09043049758795972142599547027, −8.70194313110743535508206133087, −8.005635794857326296425002843903, −7.036065015304625351490380889866, −5.492640229963999633838625601758, −4.534829212215217230154148209875, −3.35961437972062399956356609965, −3.02543030446116232331441628061, −1.7993166673430103460815846984, −0.72199491132140321723337923872, 1.145545225057148710175778035130, 2.39697516116349669573865503785, 3.68330085857688876695659488398, 4.137926361286499205081362731351, 5.54337807280782587454623205379, 6.829791639410111766863116492645, 7.18571983039049350983598113819, 7.868991629459910785608201150939, 8.83324879598233795093343568970, 9.62560742948866689489852615196, 10.51780035723893277663090102271, 10.952366491454007246453409943074, 12.710705030444391646728579511083, 13.39888654736713887594234771432, 14.1714314890850750793637843657, 14.85345239538599292296713605862, 15.63914649782441009260633062366, 16.0058203628003105664626360095, 17.206542091790983591523217568896, 18.06849357605460337807611405899, 18.7505296501933203309997610869, 19.351611892953011563493657178675, 20.16595196120006812722142552893, 20.701928922289635552390259386215, 22.15454624269161366580663435187

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