Properties

Degree 1
Conductor 569
Sign $0.999 + 0.0425i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.580 + 0.814i)2-s + (−0.251 + 0.967i)3-s + (−0.325 + 0.945i)4-s + (0.996 − 0.0883i)5-s + (−0.934 + 0.356i)6-s + (−0.997 + 0.0663i)7-s + (−0.958 + 0.283i)8-s + (−0.873 − 0.487i)9-s + (0.650 + 0.759i)10-s + (0.948 − 0.315i)11-s + (−0.833 − 0.553i)12-s + (0.970 + 0.240i)13-s + (−0.633 − 0.773i)14-s + (−0.165 + 0.986i)15-s + (−0.787 − 0.616i)16-s + (−0.219 − 0.975i)17-s + ⋯
L(s,χ)  = 1  + (0.580 + 0.814i)2-s + (−0.251 + 0.967i)3-s + (−0.325 + 0.945i)4-s + (0.996 − 0.0883i)5-s + (−0.934 + 0.356i)6-s + (−0.997 + 0.0663i)7-s + (−0.958 + 0.283i)8-s + (−0.873 − 0.487i)9-s + (0.650 + 0.759i)10-s + (0.948 − 0.315i)11-s + (−0.833 − 0.553i)12-s + (0.970 + 0.240i)13-s + (−0.633 − 0.773i)14-s + (−0.165 + 0.986i)15-s + (−0.787 − 0.616i)16-s + (−0.219 − 0.975i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.999 + 0.0425i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.999 + 0.0425i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(569\)
\( \varepsilon \)  =  $0.999 + 0.0425i$
motivic weight  =  \(0\)
character  :  $\chi_{569} (42, \cdot )$
Sato-Tate  :  $\mu(568)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 569,\ (1:\ ),\ 0.999 + 0.0425i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.434664801 + 0.03055087413i$
$L(\frac12,\chi)$  $\approx$  $1.434664801 + 0.03055087413i$
$L(\chi,1)$  $\approx$  0.9844076221 + 0.6694334245i
$L(1,\chi)$  $\approx$  0.9844076221 + 0.6694334245i

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

−22.9816878283976865860919085789, −22.14777403432059828694611943232, −21.67556251837557024113010846902, −20.35163552385699868725075230127, −19.8317989862134625579196117562, −18.876833224642061816302937192153, −18.30015061657222480807792021703, −17.36588159836651013567186880746, −16.54186166877757615190044122552, −15.12997924547366839767546241862, −14.13333091781528822388431567516, −13.57239330850798245475997360274, −12.6458975502125867062716474064, −12.38285752250501398228503190331, −11.00635309860933436572534239187, −10.37567481760986222240715100741, −9.33054194880245818117323329383, −8.467892930825585319960248701821, −6.643777160514648628739768961294, −6.31716853569864474259657367879, −5.54375520906047476190922293118, −4.05198845281597807843259099117, −3.041194698765087050617996587, −1.87205077186367878681663346783, −1.2566428125290985822076636286, 0.29164824278668903594227653189, 2.43330345767962431485032553904, 3.60384830874173621667613983561, 4.288205675371200533126661637805, 5.53577912748938037447360966986, 6.1663572604555704389229260906, 6.75662362277490605066394502938, 8.46297873511159054363231038740, 9.26386615839619506337558700984, 9.76384485276816704324010605019, 11.12072754973293951938710346975, 11.98970243569585093312265638254, 13.235555725734737493020349835351, 13.724343953050517494653160801120, 14.67930848073770549157143351027, 15.54427988704982579884425841284, 16.41110147090311644356996752483, 16.79406567710520466897048571532, 17.69120095168152297420358508867, 18.638207287947718683193295885583, 20.06660197195259521763714595604, 20.86497385901156873538143535260, 21.76745687470226061818369269438, 22.15578394826220858387449806833, 22.88384814871212497862010729902

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