Properties

Degree 1
Conductor $ 2^{5} \cdot 5^{2} $
Sign $-0.701 - 0.712i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.156 − 0.987i)3-s + 7-s + (−0.951 + 0.309i)9-s + (0.891 − 0.453i)11-s + (0.453 − 0.891i)13-s + (−0.587 − 0.809i)17-s + (−0.156 + 0.987i)19-s + (−0.156 − 0.987i)21-s + (0.309 − 0.951i)23-s + (0.453 + 0.891i)27-s + (0.987 − 0.156i)29-s + (−0.809 + 0.587i)31-s + (−0.587 − 0.809i)33-s + (−0.891 − 0.453i)37-s + (−0.951 − 0.309i)39-s + ⋯
L(s,χ)  = 1  + (−0.156 − 0.987i)3-s + 7-s + (−0.951 + 0.309i)9-s + (0.891 − 0.453i)11-s + (0.453 − 0.891i)13-s + (−0.587 − 0.809i)17-s + (−0.156 + 0.987i)19-s + (−0.156 − 0.987i)21-s + (0.309 − 0.951i)23-s + (0.453 + 0.891i)27-s + (0.987 − 0.156i)29-s + (−0.809 + 0.587i)31-s + (−0.587 − 0.809i)33-s + (−0.891 − 0.453i)37-s + (−0.951 − 0.309i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.701 - 0.712i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.701 - 0.712i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(800\)    =    \(2^{5} \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.701 - 0.712i$
motivic weight  =  \(0\)
character  :  $\chi_{800} (373, \cdot )$
Sato-Tate  :  $\mu(40)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 800,\ (1:\ ),\ -0.701 - 0.712i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8086519731 - 1.930777958i$
$L(\frac12,\chi)$  $\approx$  $0.8086519731 - 1.930777958i$
$L(\chi,1)$  $\approx$  1.034078380 - 0.5559566358i
$L(1,\chi)$  $\approx$  1.034078380 - 0.5559566358i

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

−22.04667230759524601876254941186, −21.6521242811852499701613366646, −20.91050653455630049052765693321, −20.0460228722509676013187025382, −19.40067614653091673359904854761, −18.13828908466644350744063138857, −17.29607310269466633340366504524, −16.94900581409592757842047552001, −15.720464824527609232223701058385, −15.1787818414898932233613004938, −14.35568813947170326502478562763, −13.67667514149724526527820338474, −12.34151652119272301346156758836, −11.33632456369380639938698431232, −11.09290264736032497987576068379, −9.9387455306750347504621463868, −9.00090037337994931929131488047, −8.543997663247256104498140130871, −7.189059895802187621519556174245, −6.256798077156142853975660825758, −5.17411512183442536222643567259, −4.36008410042212799369734276113, −3.75460805410632306656386029168, −2.30459007600331450494928644356, −1.210638047292727027278949583491, 0.51741744334426752752913859706, 1.38668284153785974643875175120, 2.39469324081084914412733260866, 3.56561294026189059907370982296, 4.8313652561972862869709240825, 5.74387000811423036894388783737, 6.59968993012330733035742028405, 7.50113848363986511286024296548, 8.39311190259911325621715875885, 8.91422670555284423638684897407, 10.48183136329240413467864803010, 11.151153423200121121190595104302, 11.99234367376281964858514810509, 12.6429305707257862920098172927, 13.77301717649673156436906208287, 14.216964868523150117096732316619, 15.10614037249190148451884491702, 16.322436569568673887896832682498, 17.06729270513051582360182221984, 17.99344191949995363041512227241, 18.29571598554301180734265967285, 19.35428167168075967014911476027, 20.110811940222994962971463518018, 20.82904156467465657992007690578, 21.83684679928582590188244996557

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