Properties

Degree 1
Conductor 101
Sign $0.913 + 0.406i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.309 − 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + 6-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + 10-s + (−0.809 + 0.587i)11-s + (0.309 − 0.951i)12-s + (0.309 − 0.951i)13-s + 14-s + (−0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + 17-s + ⋯
L(s,χ)  = 1  + (0.309 − 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 − 0.587i)4-s + (0.309 + 0.951i)5-s + 6-s + (0.309 + 0.951i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + 10-s + (−0.809 + 0.587i)11-s + (0.309 − 0.951i)12-s + (0.309 − 0.951i)13-s + 14-s + (−0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + 17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.913 + 0.406i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.913 + 0.406i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(101\)
\( \varepsilon \)  =  $0.913 + 0.406i$
motivic weight  =  \(0\)
character  :  $\chi_{101} (84, \cdot )$
Sato-Tate  :  $\mu(5)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 101,\ (0:\ ),\ 0.913 + 0.406i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.170300398 + 0.2484385405i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.170300398 + 0.2484385405i\)
\(L(\chi,1)\)  \(\approx\)  \(1.213473056 + 0.05008439967i\)
\(L(1,\chi)\)  \(\approx\)  \(1.213473056 + 0.05008439967i\)

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

−29.87620175434498068675722650546, −29.05197035514916823333942618866, −27.58432158030991082025459322339, −26.3542675731647677982998404983, −25.53302175650120354288368258867, −24.50510127369395816930649794236, −23.72043353292465888555584384061, −23.237078440033615418448459083037, −21.34882122430131181024677338210, −20.57389622393236254293260230852, −19.06260599883350819428205565478, −18.011478517743716218000011403262, −16.90852810903864632338590304486, −16.250393458663906533142775676115, −14.48860550409057274476577392708, −13.66192481285689082680881346675, −12.99203695053520790697848827811, −11.73219341950956844047685435940, −9.635556568687012088472503073686, −8.29895243302748370050690090822, −7.642610502334181228000906996659, −6.2576938267698971749764155035, −5.09543531747327189959110991111, −3.5358446117796721455320123600, −1.24579123951389486967156515188, 2.44721625696651235472626437450, 3.13847628243686989161624626094, 4.83151864812607189909974953418, 5.80185667505013925215786195683, 8.0965103601528089956776476951, 9.4407338783349414660916767570, 10.36866350110202421976480001728, 11.16592547469801879943249808355, 12.54690543629199417502796708609, 13.93181913363197132862936562730, 14.91131410399127854159114538800, 15.590738296272601811126617759994, 17.65648610037278988327969632888, 18.456952405189809032225188573116, 19.620951471228520162545972515193, 20.888212914121227815859652069477, 21.4203086683588974765130578912, 22.46842413990106402050849469310, 23.1305562915006858384308198831, 25.00144736241010803147198251479, 26.08057886679188820838379962877, 27.042123944678512326345494488473, 28.07946364333104692089147440093, 28.725556036927406102794550293329, 30.2859359906572880424127981842

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