Properties

Degree 1
Conductor 71
Sign $-0.999 + 0.00177i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.858 + 0.512i)2-s + (−0.691 + 0.722i)3-s + (0.473 + 0.880i)4-s + (−0.809 + 0.587i)5-s + (−0.963 + 0.266i)6-s + (0.995 + 0.0896i)7-s + (−0.0448 + 0.998i)8-s + (−0.0448 − 0.998i)9-s + (−0.995 + 0.0896i)10-s + (−0.936 + 0.351i)11-s + (−0.963 − 0.266i)12-s + (−0.936 − 0.351i)13-s + (0.809 + 0.587i)14-s + (0.134 − 0.990i)15-s + (−0.550 + 0.834i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s,χ)  = 1  + (0.858 + 0.512i)2-s + (−0.691 + 0.722i)3-s + (0.473 + 0.880i)4-s + (−0.809 + 0.587i)5-s + (−0.963 + 0.266i)6-s + (0.995 + 0.0896i)7-s + (−0.0448 + 0.998i)8-s + (−0.0448 − 0.998i)9-s + (−0.995 + 0.0896i)10-s + (−0.936 + 0.351i)11-s + (−0.963 − 0.266i)12-s + (−0.936 − 0.351i)13-s + (0.809 + 0.587i)14-s + (0.134 − 0.990i)15-s + (−0.550 + 0.834i)16-s + (−0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.999 + 0.00177i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.999 + 0.00177i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $-0.999 + 0.00177i$
motivic weight  =  \(0\)
character  :  $\chi_{71} (7, \cdot )$
Sato-Tate  :  $\mu(70)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 71,\ (1:\ ),\ -0.999 + 0.00177i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.001297973288 + 1.463713094i$
$L(\frac12,\chi)$  $\approx$  $0.001297973288 + 1.463713094i$
$L(\chi,1)$  $\approx$  0.7957045419 + 0.8634934866i
$L(1,\chi)$  $\approx$  0.7957045419 + 0.8634934866i

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

−30.862309257730118723753977045060, −29.99926694836157913548542995771, −28.69844510743280535129827393408, −28.17598316863915660985857089802, −26.83798225579101758582560182099, −24.64953614092332023139092443615, −24.0258420165539805014732974405, −23.46912024735067128248195581065, −22.12810108609909051069807069328, −21.03254312948549589669208275861, −19.78399620168998019927063675027, −18.87745718380299933751641676304, −17.46260428423915021933912463745, −16.104974662084349574006257438479, −14.83682804563374442599083116711, −13.41801406789758470760639744121, −12.430334260534708805540461955692, −11.51282018344417786479176432890, −10.600275705545732710649822081, −8.28161532185448016585419164730, −6.98104999287732286634744498801, −5.29115255687814846053872947220, −4.4808625866327722393671741019, −2.270886921230317601226048435459, −0.59913463222098977603554087031, 2.96843453407237996636391651073, 4.51791269508967166046511580658, 5.32206484663810631871987130734, 7.02624794977909596484590909867, 8.11436346552599974766278545110, 10.301753480417092433075576338041, 11.511918433663419206487672339769, 12.236552678315350148423136089495, 14.13121461944100132756712561116, 15.21814632917579844629157011429, 15.791047373340366911772734864814, 17.20864361645630257330572707583, 18.21211167936059497377188038080, 20.24801705798925440621994411637, 21.21959933610633944809448998059, 22.271246012311076992497482909218, 23.18828563524809305500907438331, 23.904982555367418268163675134883, 25.297711371189861261991506301238, 26.9280604428416406748804928342, 27.11239964360698114573575023478, 28.85264183321392945077832534088, 29.997886376751214071051770993609, 31.242240722851148627115941711700, 31.807174147970593640105262088901

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