Properties

Degree 1
Conductor $ 2^{2} \cdot 5^{2} $
Sign $-0.425 - 0.904i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.425 - 0.904i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.425 - 0.904i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(100\)    =    \(2^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.425 - 0.904i$
motivic weight  =  \(0\)
character  :  $\chi_{100} (91, \cdot )$
Sato-Tate  :  $\mu(10)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 100,\ (1:\ ),\ -0.425 - 0.904i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.8766659312 - 1.381404465i$
$L(\frac12,\chi)$  $\approx$  $0.8766659312 - 1.381404465i$
$L(\chi,1)$  $\approx$  1.081388548 - 0.5088628249i
$L(1,\chi)$  $\approx$  1.081388548 - 0.5088628249i

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

−30.24615714896790073330705470208, −28.74431315818797254959667785543, −28.08373105489283670368226275825, −26.53137720985088388777766558092, −26.08803405214710040190901032169, −25.12369461741461590002489448169, −23.81812575852650052871137352290, −22.50956893417377456953233178968, −21.65999650981566356339488933406, −20.4480744630058923183436069852, −19.65527082418544219971175256652, −18.63529763095557945643152890254, −17.10740479751194009774167731158, −15.84984227122171565308816471233, −15.22930769967971878173273303110, −13.78784386483765987010101211287, −12.980353061389130288440278164480, −11.37097444406704119480845023184, −9.868203450849767536774734686318, −9.30204804026100103237633552484, −7.828921442575390296806767069154, −6.507893844168187548348781212775, −4.70779111615043734242485773058, −3.520180667568283749031222535, −2.11339860590120723361623431921, 0.64137602175394656600372314701, 2.63975559294522670082553493352, 3.6278295194698934810972369339, 5.76120343778232217872878691512, 6.97226152605682535644106722173, 8.231898837148160993256276256272, 9.26916468857723266957295390019, 10.58174974812249823811777080684, 12.23153643508396595840924423055, 13.22079360501403320305683916530, 14.02133146096858480011795136366, 15.44460274203615558660128542124, 16.34985778180431588617065991601, 18.02695019581774911426945875832, 18.80243520614971783169961517363, 19.88385278796923605576659389532, 20.66149888086729297366626143673, 22.12063142293760181826736299106, 23.1418415768954438213430350305, 24.43765355289426877635143857338, 25.12645808754741957099875940935, 26.262641828281791479616076171637, 26.94789320567263350275019167036, 28.61410971977832494221238475951, 29.460700846449877005306814306095

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