Properties

Degree 1
Conductor 71
Sign $-0.276 + 0.961i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)3-s + (−0.222 + 0.974i)4-s + 5-s + (−0.900 + 0.433i)6-s + (0.623 − 0.781i)7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.623 + 0.781i)10-s + (−0.900 − 0.433i)11-s + (−0.900 − 0.433i)12-s + (−0.900 + 0.433i)13-s + 14-s + (−0.222 + 0.974i)15-s + (−0.900 − 0.433i)16-s + 17-s + ⋯
L(s,χ)  = 1  + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)3-s + (−0.222 + 0.974i)4-s + 5-s + (−0.900 + 0.433i)6-s + (0.623 − 0.781i)7-s + (−0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (0.623 + 0.781i)10-s + (−0.900 − 0.433i)11-s + (−0.900 − 0.433i)12-s + (−0.900 + 0.433i)13-s + 14-s + (−0.222 + 0.974i)15-s + (−0.900 − 0.433i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.276 + 0.961i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.276 + 0.961i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(71\)
\( \varepsilon \)  =  $-0.276 + 0.961i$
motivic weight  =  \(0\)
character  :  $\chi_{71} (30, \cdot )$
Sato-Tate  :  $\mu(7)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 71,\ (0:\ ),\ -0.276 + 0.961i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7612747889 + 1.011053489i$
$L(\frac12,\chi)$  $\approx$  $0.7612747889 + 1.011053489i$
$L(\chi,1)$  $\approx$  1.045743164 + 0.8386232235i
$L(1,\chi)$  $\approx$  1.045743164 + 0.8386232235i

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

−31.238688076522595238163720414978, −30.17611993711121095620708849512, −29.32671354386646642008635972448, −28.60381218189835461716214937628, −27.50006072515652479629063134534, −25.463474799360877448483240064700, −24.70983323709063478910089555261, −23.62178088040715687937659822837, −22.53104544105962119039552750293, −21.40537061390291616937730192602, −20.51981148882355472716778396666, −18.97471835682822112836286462747, −18.25271130542732601316461359604, −17.21062043072803991383659600756, −15.0199073893692647792665128317, −14.058001889833722686339541092970, −12.852002150869048958076365941941, −12.17043626313264743665137975616, −10.761593542827476960143034066800, −9.47539591907072080219520196268, −7.70415979170948698629270109708, −5.82678321428887856735771097560, −5.23112406191882072770166633655, −2.66924559307171264058238290045, −1.71926766548892730211603334247, 2.95829288112889641250855143897, 4.692611101263191596982828670580, 5.412203147810498355168419398009, 6.96681023668174066181785253309, 8.56913050332369307347200414642, 9.960284725352793654370796862209, 11.19922513533497087883927397511, 12.9190999644571623632670515501, 14.17314315945349609986946167897, 14.85939415797528554561937936444, 16.44348671436128730267054218816, 17.0047904779714534636117572136, 18.11125483981824612277278286725, 20.42670876935574441484764762635, 21.340967603426242077926811239429, 21.95411797269092708398647449730, 23.309797228095371466158477664365, 24.210611127697574909989302490202, 25.635608743540074442902063490005, 26.42843108025866131779306159544, 27.30593989333652283951699691300, 28.89609310991235125987601295921, 29.90760296594414421841904310434, 31.27861283051213008695559945318, 32.39914539866972506873467329900

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