Properties

Degree 1
Conductor 101
Sign $-0.956 + 0.291i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.844 + 0.535i)2-s + (−0.248 + 0.968i)3-s + (0.425 − 0.904i)4-s + (0.535 − 0.844i)5-s + (−0.309 − 0.951i)6-s + (−0.368 − 0.929i)7-s + (0.125 + 0.992i)8-s + (−0.876 − 0.481i)9-s + i·10-s + (−0.481 + 0.876i)11-s + (0.770 + 0.637i)12-s + (0.929 + 0.368i)13-s + (0.809 + 0.587i)14-s + (0.684 + 0.728i)15-s + (−0.637 − 0.770i)16-s + (−0.309 + 0.951i)17-s + ⋯
L(s,χ)  = 1  + (−0.844 + 0.535i)2-s + (−0.248 + 0.968i)3-s + (0.425 − 0.904i)4-s + (0.535 − 0.844i)5-s + (−0.309 − 0.951i)6-s + (−0.368 − 0.929i)7-s + (0.125 + 0.992i)8-s + (−0.876 − 0.481i)9-s + i·10-s + (−0.481 + 0.876i)11-s + (0.770 + 0.637i)12-s + (0.929 + 0.368i)13-s + (0.809 + 0.587i)14-s + (0.684 + 0.728i)15-s + (−0.637 − 0.770i)16-s + (−0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.956 + 0.291i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.956 + 0.291i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(101\)
\( \varepsilon \)  =  $-0.956 + 0.291i$
motivic weight  =  \(0\)
character  :  $\chi_{101} (72, \cdot )$
Sato-Tate  :  $\mu(100)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 101,\ (1:\ ),\ -0.956 + 0.291i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.08030581240 + 0.5392848313i$
$L(\frac12,\chi)$  $\approx$  $0.08030581240 + 0.5392848313i$
$L(\chi,1)$  $\approx$  0.5292342760 + 0.2811085605i
$L(1,\chi)$  $\approx$  0.5292342760 + 0.2811085605i

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

−28.99786953058447688315505082501, −28.56950448501124751418004660181, −27.185204112186613650199402175558, −25.87941503471850874472640430425, −25.38240850211121046780871275463, −24.25546760913235575794554161570, −22.656925862181712137613007813252, −21.88451514227733155302877009257, −20.64898566553416300247352916382, −19.21065689861953168270929708404, −18.49640155998386930072467809722, −18.05090259392117806307546891796, −16.72702295907687488542492192130, −15.44595477030299204701617157528, −13.6608495825774142462788867277, −12.83077697549538741447803459891, −11.459137710432503944136921441198, −10.76651713865513919544647895278, −9.19747617812273692968233683739, −8.139845071533429593628834294941, −6.76041919690064613598880305486, −5.85902537215085152814131322261, −3.012172338416597457532756625433, −2.20125977038519455888702182448, −0.32023234531304627760826658075, 1.54064684616581696805904270206, 4.02886269041131843544147238267, 5.349826881783232305301816825001, 6.49798737795101469645369812627, 8.14983349315534599349203248359, 9.301395592897430390188422353687, 10.13643248074209836450580709478, 11.04760651440142665444550924597, 12.88778560700578362976047304203, 14.29615990180023607512919371213, 15.52596476240677146420020467112, 16.49378379768441551688510446800, 17.0887963228754057899920016475, 18.10967693879559163202935353091, 19.84453760142926517251772036842, 20.48209865370487182035212535595, 21.52123748160336861546783549970, 23.2503187546342673230515964358, 23.71224607805474340571640009060, 25.508294891057768622391842456083, 25.85784622855872600294794272321, 27.03645599854117920461431769558, 28.056383023128138509305792588755, 28.64576946822781387364277602859, 29.62063024183482259973321658305

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