Properties

Degree 1
Conductor 101
Sign $0.421 - 0.906i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.368 − 0.929i)2-s + (−0.770 − 0.637i)3-s + (−0.728 + 0.684i)4-s + (−0.929 − 0.368i)5-s + (−0.309 + 0.951i)6-s + (0.248 + 0.968i)7-s + (0.904 + 0.425i)8-s + (0.187 + 0.982i)9-s + i·10-s + (0.982 − 0.187i)11-s + (0.998 − 0.0627i)12-s + (−0.968 − 0.248i)13-s + (0.809 − 0.587i)14-s + (0.481 + 0.876i)15-s + (0.0627 − 0.998i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s,χ)  = 1  + (−0.368 − 0.929i)2-s + (−0.770 − 0.637i)3-s + (−0.728 + 0.684i)4-s + (−0.929 − 0.368i)5-s + (−0.309 + 0.951i)6-s + (0.248 + 0.968i)7-s + (0.904 + 0.425i)8-s + (0.187 + 0.982i)9-s + i·10-s + (0.982 − 0.187i)11-s + (0.998 − 0.0627i)12-s + (−0.968 − 0.248i)13-s + (0.809 − 0.587i)14-s + (0.481 + 0.876i)15-s + (0.0627 − 0.998i)16-s + (−0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.421 - 0.906i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.421 - 0.906i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(101\)
\( \varepsilon \)  =  $0.421 - 0.906i$
motivic weight  =  \(0\)
character  :  $\chi_{101} (3, \cdot )$
Sato-Tate  :  $\mu(100)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 101,\ (1:\ ),\ 0.421 - 0.906i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6265063679 - 0.3996794898i$
$L(\frac12,\chi)$  $\approx$  $0.6265063679 - 0.3996794898i$
$L(\chi,1)$  $\approx$  0.5361099544 - 0.2895516876i
$L(1,\chi)$  $\approx$  0.5361099544 - 0.2895516876i

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

−29.76331897817255820406244210464, −28.32565967679338416904448910252, −27.54062683686528370516444240295, −26.7300305538297744148828675086, −26.15865388380552960988694764103, −24.29214466767655972693952510342, −23.738918445518445645216753488145, −22.66575672684645984528255326527, −21.9960419341338061458751697220, −20.10116766413048208520547961520, −19.24956463929752183264168747593, −17.682670815818231136547072223616, −17.05881723854953285448088162685, −16.06639862431293009175835276137, −15.02405532930693315096049103249, −14.21640281363606702817897943111, −12.32138335425406548342328029351, −10.99447177836538575384801466503, −10.117015218139323086014858328043, −8.73533227856063061500814307210, −7.23676880928293358885851047845, −6.505330601412869122166466596925, −4.70641289472066855591690129105, −4.02288427313489577555909102003, −0.68853693257504646576266964685, 0.78427473699542099157769068894, 2.35219996784977520852111901796, 4.18372873101104936264627520718, 5.48727917544509992603883929849, 7.36353071228879976269788427625, 8.38919773140465037651665740307, 9.70618619136353638811190420282, 11.37311968918482111075016816456, 11.91176744035460071208163593835, 12.58063434043903324030121008280, 14.1156344663327889150667375676, 15.83859010839865630181612124787, 17.020223361118152265579604084913, 17.94931444075290777367207816723, 19.097334715723292245590327446543, 19.61086478349535011860392283213, 21.03826920147123890798329126942, 22.315774338389892585045667840460, 22.836177321512664055067981431734, 24.33684400151993562963797268052, 25.06331719733095816097564802614, 26.99804334564512070019269808005, 27.60367285116527978167136815081, 28.392497456373057183421891704318, 29.41564242006042524380267382898

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