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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(32\)    =    \(2^{5}\)
\( \varepsilon \)  =  $0.555 + 0.831i$
motivic weight  =  \(0\)
character  :  $\chi_{32} (5, \cdot )$
Sato-Tate  :  $\mu(8)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 32,\ (0:\ ),\ 0.555 + 0.831i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5968585698 + 0.3190275521i$
$L(\frac12,\chi)$  $\approx$  $0.5968585698 + 0.3190275521i$
$L(\chi,1)$  $\approx$  0.8128533770 + 0.2721024297i
$L(1,\chi)$  $\approx$  0.8128533770 + 0.2721024297i

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

−36.04287367230733825865025619868, −35.635484288982439277980201306356, −33.60711845578218687389313168416, −33.271760896971241898260960317222, −31.38069032392039881602813044947, −30.06287910870776622604982137380, −28.95924076086137830462881964731, −28.22046986633945729073296017888, −26.3958271755062273420650548793, −24.99402323434118102820059668551, −23.8420931815044262475601627185, −22.88159429767618842563350980403, −21.20792858558038908661694309476, −19.94902534701121500851990746527, −18.21917751697834170267040614454, −17.257161714752641343305979360589, −16.1320222254487730501484775984, −13.74066443950421262090986180363, −12.978060800643753945329949705474, −11.34417060495042936727964131282, −9.81781071538947725499538815108, −7.78989884395503367104593862974, −6.27178192155914422380013490473, −4.68540358759759307267341170842, −1.61796366745704267360678474775, 2.93980914934491876882348312166, 5.26830475631644288871154228503, 6.36900691647196385487732509292, 8.840836184286508683008535274278, 10.35751758728224776296900481828, 11.41447865465362867125541895181, 13.23633554137878311828454913452, 15.022478055371743092833375428495, 16.02997273108670983286583311768, 17.73970088705665045545356340097, 18.52915373087765196114113297930, 20.72886352148557734263381799325, 21.87840323758562610190290265399, 22.5790061497568081238836339054, 24.27706967405060115307228629307, 25.81909098311021865364130611181, 26.87257388047438987945123649392, 28.34579119564014013345931114692, 29.11198174562442421990740925592, 30.6179077939016069215284902426, 32.15736094397393362013552506267, 33.25622908968484106913004639477, 34.33873934094434389032606276883, 35.11950716231660457387726539327, 37.27797253488685759462698003677

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