Properties

Degree 1
Conductor 569
Sign $0.640 + 0.768i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.730 − 0.683i)2-s + (−0.408 + 0.912i)3-s + (0.0663 + 0.997i)4-s + (0.325 − 0.945i)5-s + (0.921 − 0.387i)6-s + (−0.598 + 0.801i)7-s + (0.633 − 0.773i)8-s + (−0.666 − 0.745i)9-s + (−0.883 + 0.467i)10-s + (0.975 + 0.219i)11-s + (−0.937 − 0.346i)12-s + (0.964 + 0.262i)13-s + (0.984 − 0.176i)14-s + (0.730 + 0.683i)15-s + (−0.991 + 0.132i)16-s + (−0.999 − 0.0442i)17-s + ⋯
L(s,χ)  = 1  + (−0.730 − 0.683i)2-s + (−0.408 + 0.912i)3-s + (0.0663 + 0.997i)4-s + (0.325 − 0.945i)5-s + (0.921 − 0.387i)6-s + (−0.598 + 0.801i)7-s + (0.633 − 0.773i)8-s + (−0.666 − 0.745i)9-s + (−0.883 + 0.467i)10-s + (0.975 + 0.219i)11-s + (−0.937 − 0.346i)12-s + (0.964 + 0.262i)13-s + (0.984 − 0.176i)14-s + (0.730 + 0.683i)15-s + (−0.991 + 0.132i)16-s + (−0.999 − 0.0442i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.640 + 0.768i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.640 + 0.768i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $0.640 + 0.768i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (20, \cdot )$
Sato-Tate  :  $\mu(142)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (0:\ ),\ 0.640 + 0.768i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6778959525 + 0.3176014369i$
$L(\frac12,\chi)$  $\approx$  $0.6778959525 + 0.3176014369i$
$L(\chi,1)$  $\approx$  0.6783993325 + 0.04991298465i
$L(1,\chi)$  $\approx$  0.6783993325 + 0.04991298465i

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

−23.24349636607043558222416281295, −22.64440560640260156737191688352, −21.837278838035239059033282177499, −19.98569251886152140037468941196, −19.73456691370365820622459460183, −18.72329879640288667838517756410, −18.04415552705026954055601975959, −17.43337373918688784839907350012, −16.64110696248796378401961022802, −15.75494821379593749620228261908, −14.65884566678557193983477939110, −13.70874672465101728849274232297, −13.38876480847707040721387135398, −11.79761179307612685411182426287, −10.8613985749029379204347605229, −10.38810148186336620564369283095, −9.11740211559484769328674569229, −8.223034330836365554927367715929, −7.08171067883730564503011742658, −6.47462581626508670913960973241, −6.20881718319437740168262639220, −4.630958351288498654246956184847, −3.0696825857799336759921830435, −1.790309817082906272054966964774, −0.62248382382359769370339139864, 1.122991767703118100979173538978, 2.35034578020695956714015842601, 3.75945096171795182436191751967, 4.318403077910886262608829087165, 5.74550222745820291471793102649, 6.47520340256366621342952881599, 8.20079734978852775536365526260, 9.06481852244186674806304296850, 9.39558639189832705182556398249, 10.29307442420754239833002158450, 11.46006032197283082353063245526, 11.98550625109851002277603659272, 12.85322899833703994661258949768, 13.8837752758191091385896090753, 15.340954865850751061800834883355, 16.125702371551976909601675091485, 16.6200195874900308891538233826, 17.5565236046963208269758347338, 18.187907046320772005631835859405, 19.46199820220114794884543578601, 20.03337726523628319025087838477, 21.00672099380872904046105068478, 21.45473260381426386511436376364, 22.30614960940336920152677149091, 22.9945322736605222986723223969

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