Properties

Degree 1
Conductor 101
Sign $-0.941 + 0.337i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.982 + 0.187i)2-s + (−0.904 + 0.425i)3-s + (0.929 − 0.368i)4-s + (−0.187 + 0.982i)5-s + (0.809 − 0.587i)6-s + (0.125 + 0.992i)7-s + (−0.844 + 0.535i)8-s + (0.637 − 0.770i)9-s i·10-s + (0.770 + 0.637i)11-s + (−0.684 + 0.728i)12-s + (0.992 + 0.125i)13-s + (−0.309 − 0.951i)14-s + (−0.248 − 0.968i)15-s + (0.728 − 0.684i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s,χ)  = 1  + (−0.982 + 0.187i)2-s + (−0.904 + 0.425i)3-s + (0.929 − 0.368i)4-s + (−0.187 + 0.982i)5-s + (0.809 − 0.587i)6-s + (0.125 + 0.992i)7-s + (−0.844 + 0.535i)8-s + (0.637 − 0.770i)9-s i·10-s + (0.770 + 0.637i)11-s + (−0.684 + 0.728i)12-s + (0.992 + 0.125i)13-s + (−0.309 − 0.951i)14-s + (−0.248 − 0.968i)15-s + (0.728 − 0.684i)16-s + (0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.941 + 0.337i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.941 + 0.337i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(101\)
\( \varepsilon \)  =  $-0.941 + 0.337i$
motivic weight  =  \(0\)
character  :  $\chi_{101} (63, \cdot )$
Sato-Tate  :  $\mu(100)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 101,\ (1:\ ),\ -0.941 + 0.337i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.1252119593 + 0.7209079804i$
$L(\frac12,\chi)$  $\approx$  $0.1252119593 + 0.7209079804i$
$L(\chi,1)$  $\approx$  0.4729930304 + 0.3554763936i
$L(1,\chi)$  $\approx$  0.4729930304 + 0.3554763936i

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

−29.154292879489599379315846374602, −27.91158179500030783221992722978, −27.60205713796196683372396996744, −26.28601596359941543315639622435, −24.89587230104861013934740362511, −24.15739637281576043092814420900, −23.19905252918668120255682901768, −21.69711201483832972291673880214, −20.47490324287405453355553039016, −19.69236421802572952974541934008, −18.42296896128008323071448890602, −17.4888988386021637500355614656, −16.428928393550015947908597591377, −16.15126371553559513000365859250, −13.83573288677692979468567460053, −12.56177435189002409434463604272, −11.54534542530792008993094365608, −10.680236193878542459409797406426, −9.27886697728035020246422299005, −8.00828280640849472768298333186, −6.94523943453239914499352224023, −5.5829502335369038192606417163, −3.82907592728410702492876090734, −1.34420790701120858288573310584, −0.577510190587569467842655238345, 1.63741330143079935155928702782, 3.58073643819869342066506904411, 5.72584128348305858375228643367, 6.46586510334847381726650913293, 7.858927287705547134578521446778, 9.40197629567495547955185638806, 10.30185235930578220786928187521, 11.4893515072725365708136741256, 12.083842093136669177112455967958, 14.53415015205372094039414328500, 15.40771083074179582643095370543, 16.29469815844586319603743685613, 17.59169198412759923804733058565, 18.30943013512889005523292889427, 19.14276433226020458387789145627, 20.69225968645783706322166527637, 21.792941248563544167200538316332, 22.81002969264174427787964006606, 23.87521574377781752593815846566, 25.25107618563321952818179918973, 26.05542804017471976435171315234, 27.29288480187477957181510659247, 27.879083927375173936996750811644, 28.7272554192674939902620469267, 29.88773350289455021637518499688

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