Properties

Degree 1
Conductor $ 2^{5} $
Sign $-0.555 - 0.831i$
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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(32\)    =    \(2^{5}\)
\( \varepsilon \)  =  $-0.555 - 0.831i$
motivic weight  =  \(0\)
character  :  $\chi_{32} (11, \cdot )$
Sato-Tate  :  $\mu(8)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 32,\ (1:\ ),\ -0.555 - 0.831i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.2157838221 - 0.4037031366i$
$L(\frac12,\chi)$  $\approx$  $0.2157838221 - 0.4037031366i$
$L(\chi,1)$  $\approx$  0.5895674212 - 0.1172722514i
$L(1,\chi)$  $\approx$  0.5895674212 - 0.1172722514i

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

−36.563320831480398187828733150276, −35.11018580848094682067979262108, −34.595983483177342598268965004987, −33.42615366434263286749973258700, −31.39405175047235851176092492747, −30.71386947741491043347034624890, −29.27947007484902521818795674033, −28.2262756654867517525883862310, −26.97053742034207476158609514880, −25.3268854178902836268812067947, −24.15325091339672301916403862563, −22.83135187919236460034550904823, −22.07061394708290993310184347701, −19.97480755039527644488342556301, −18.579847968191037089237283116568, −17.88874729907091804151002217436, −16.01486636107363878259656233839, −14.78330839875493402363799741298, −12.78239739432504252037980505139, −11.80290245042927264818206101295, −10.401582824759769374713253267141, −8.08103896471699337191156739253, −6.77857064340597392951572332309, −5.12118384634175354038228292470, −2.52335569027335173359802788495, 0.335997693309994745288286596145, 3.90396140808436788368909584674, 5.15648920368548608009817582684, 7.237171006330304127500147324182, 9.11566793887460192425536653720, 10.7279912516477488942001281778, 11.85753765983173802371176480683, 13.54817089184996679921492779725, 15.51481202317953046138317638911, 16.4502156222690426019925636176, 17.55455700881077178097288024770, 19.54007392555487010952974071140, 20.71492265311507038096430588898, 22.00774425634167274349368024193, 23.487689357666505901027686275079, 24.13909475814481445734606320200, 26.48108629740884827349656064020, 27.08141175091534323190065024537, 28.508123922038415751517514977483, 29.38119164680306878354992438225, 31.2178771080785969480708569384, 32.34719867997491334363745730981, 33.40493274860043366950913642966, 34.603647099619579339330344930451, 35.76899755798925873865168627354

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