Properties

Degree 1
Conductor $ 2^{5} $
Sign $0.831 - 0.555i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.707 − 0.707i)3-s + (0.707 + 0.707i)5-s i·7-s i·9-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s + 15-s − 17-s + (−0.707 + 0.707i)19-s + (−0.707 − 0.707i)21-s + i·23-s + i·25-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)29-s − 31-s + ⋯
L(s,χ)  = 1  + (0.707 − 0.707i)3-s + (0.707 + 0.707i)5-s i·7-s i·9-s + (0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s + 15-s − 17-s + (−0.707 + 0.707i)19-s + (−0.707 − 0.707i)21-s + i·23-s + i·25-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)29-s − 31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.831 - 0.555i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 32 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.831 - 0.555i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(32\)    =    \(2^{5}\)
\( \varepsilon \)  =  $0.831 - 0.555i$
motivic weight  =  \(0\)
character  :  $\chi_{32} (27, \cdot )$
Sato-Tate  :  $\mu(8)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 32,\ (1:\ ),\ 0.831 - 0.555i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.808198019 - 0.5485108726i$
$L(\frac12,\chi)$  $\approx$  $1.808198019 - 0.5485108726i$
$L(\chi,1)$  $\approx$  1.423341664 - 0.2831202600i
$L(1,\chi)$  $\approx$  1.423341664 - 0.2831202600i

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

−36.64985030842321296223373644535, −35.31581570894988921655710203111, −33.69864573992814775165743979532, −32.58925639882058419378699304750, −31.77901428437357431018605218581, −30.50889164782938318855510252930, −28.73683384048286617104121249941, −27.85937665149967744962289924955, −26.38109685862421843718963220524, −25.21821206629074999356042207656, −24.31756691525862003410131205072, −22.038890501729185440827949094937, −21.368882435449109982458532901573, −20.09281129587004903401079664786, −18.68296180803459633844611346848, −16.89273313282052864794273354121, −15.72094109915403842745920173932, −14.29611150719206453655236933967, −13.028204813562807670242051480390, −11.12231938416884050823641586706, −9.193735552677570762148667626089, −8.71610959747903301666137937837, −6.032042852509231921086069214510, −4.33888362925581802874904501208, −2.24335513554469438713146152642, 1.72096909693814813092436307440, 3.640656916871261635983272091893, 6.35900187234094306006619502021, 7.52319196293555974166849458647, 9.33658997625249447827700621129, 10.84912514697350119980684825974, 12.89527440790653569427679489166, 13.92789197325934873393141496216, 15.04019301381373658472390381402, 17.26187013802039644735136478376, 18.234247045831694711022588586676, 19.70766320514182973203812787312, 20.72569155984747047415154155781, 22.47027247730529576759225056746, 23.70308022984907141220013318767, 25.26204011432168976313088623105, 25.92210250981586179255059257165, 27.30201585136737243001613384566, 29.28612035597456373320287123621, 30.057899216613783309087949166571, 30.97942908874554956522548170408, 32.6668073754382393468188834826, 33.55058988425371830514389971908, 35.26702705750753035320270543181, 36.191648219598528379062050628373

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