Properties

Degree 1
Conductor $ 2^{6} \cdot 5 $
Sign $-0.502 + 0.864i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.382 + 0.923i)3-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s + (0.923 + 0.382i)13-s + 17-s + (−0.382 + 0.923i)19-s + (−0.923 + 0.382i)21-s + (−0.707 + 0.707i)23-s + (0.923 − 0.382i)27-s + (−0.923 − 0.382i)29-s + 31-s i·33-s + (−0.923 + 0.382i)37-s + (−0.707 + 0.707i)39-s + ⋯
L(s,χ)  = 1  + (−0.382 + 0.923i)3-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s + (0.923 + 0.382i)13-s + 17-s + (−0.382 + 0.923i)19-s + (−0.923 + 0.382i)21-s + (−0.707 + 0.707i)23-s + (0.923 − 0.382i)27-s + (−0.923 − 0.382i)29-s + 31-s i·33-s + (−0.923 + 0.382i)37-s + (−0.707 + 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.502 + 0.864i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 320 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.502 + 0.864i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(320\)    =    \(2^{6} \cdot 5\)
\( \varepsilon \)  =  $-0.502 + 0.864i$
motivic weight  =  \(0\)
character  :  $\chi_{320} (3, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 320,\ (0:\ ),\ -0.502 + 0.864i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.5017637674 + 0.8718459638i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.5017637674 + 0.8718459638i\)
\(L(\chi,1)\)  \(\approx\)  \(0.8183212376 + 0.4558308855i\)
\(L(1,\chi)\)  \(\approx\)  \(0.8183212376 + 0.4558308855i\)

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

−24.62777805748939081360727916437, −23.82661104996601435362161290760, −23.36250278143740542657827439800, −22.39718480721994878752799960597, −21.116604156419735651029747844267, −20.438558299741642466502568434247, −19.31930918075323143014438492362, −18.42278365016264736531234103980, −17.78409795740037628185888057766, −16.85744673321071215234280592306, −15.95456716983012432862895129290, −14.63599497398326865167763392894, −13.64815709571221561579153825111, −13.077844192981583710738391407386, −11.9264443028240875024330698159, −10.97483655258909927462974250336, −10.32388893344471970824466944979, −8.50680537398287716754885666531, −7.91141588379183710342812399863, −6.90666929038235930111112736368, −5.79062285794098302407730690306, −4.85939107983190506330707322333, −3.32844979881181375301103282368, −1.94937688840660461366794030940, −0.692863331025744953383754183483, 1.72079154634831705871647142900, 3.21184535919321349899278745906, 4.339686313348678559563642837440, 5.40612260192459299207251559104, 6.0698667692528441506956291992, 7.783950752608875011624614164084, 8.630134606606667756240244818808, 9.77201492767842929637269624542, 10.57402587239495451553266138305, 11.58861542677078500663138802599, 12.297996795088241375369002660591, 13.718267716689510384095376821286, 14.74886966479028896309531263795, 15.5063184359915760583298226312, 16.269594987862398948096006622165, 17.29551533283879001571405354134, 18.20230637331198147140469173908, 18.98335300910701694968903262099, 20.569463592607458620489676728999, 21.00133661504798199635280615767, 21.66800114100650602172885787883, 22.87128779366435981667235797088, 23.42422164184174521182491251048, 24.497095064555669718922258897991, 25.73688411858253795024732727343

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