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  :  $\chi_{101} (95, \cdot )$
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

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

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