Properties

Degree 1
Conductor 101
Sign $0.941 - 0.338i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.248 − 0.968i)2-s + (0.998 − 0.0627i)3-s + (−0.876 + 0.481i)4-s + (0.968 + 0.248i)5-s + (−0.309 − 0.951i)6-s + (−0.770 + 0.637i)7-s + (0.684 + 0.728i)8-s + (0.992 − 0.125i)9-s i·10-s + (0.125 + 0.992i)11-s + (−0.844 + 0.535i)12-s + (0.637 − 0.770i)13-s + (0.809 + 0.587i)14-s + (0.982 + 0.187i)15-s + (0.535 − 0.844i)16-s + (−0.309 + 0.951i)17-s + ⋯
L(s,χ)  = 1  + (−0.248 − 0.968i)2-s + (0.998 − 0.0627i)3-s + (−0.876 + 0.481i)4-s + (0.968 + 0.248i)5-s + (−0.309 − 0.951i)6-s + (−0.770 + 0.637i)7-s + (0.684 + 0.728i)8-s + (0.992 − 0.125i)9-s i·10-s + (0.125 + 0.992i)11-s + (−0.844 + 0.535i)12-s + (0.637 − 0.770i)13-s + (0.809 + 0.587i)14-s + (0.982 + 0.187i)15-s + (0.535 − 0.844i)16-s + (−0.309 + 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.941 - 0.338i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.941 - 0.338i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(101\)
\( \varepsilon \)  =  $0.941 - 0.338i$
motivic weight  =  \(0\)
character  :  $\chi_{101} (12, \cdot )$
Sato-Tate  :  $\mu(100)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 101,\ (1:\ ),\ 0.941 - 0.338i)$
$L(\chi,\frac{1}{2})$  $\approx$  $2.186603091 - 0.3808328061i$
$L(\frac12,\chi)$  $\approx$  $2.186603091 - 0.3808328061i$
$L(\chi,1)$  $\approx$  1.385218825 - 0.3347323994i
$L(1,\chi)$  $\approx$  1.385218825 - 0.3347323994i

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

−29.61687557508762285681697582208, −28.67468823942813440755938270815, −27.14467838457070272959466015348, −26.31499556936712346402179522033, −25.67551509252930833632307881213, −24.70853448788848825500727894095, −23.9032737345320894135823892027, −22.43151620817726600399765002510, −21.38092385105866321030380143216, −20.11777493166186491853042521819, −19.00179899842403478263870424496, −18.07576797699778557775573720215, −16.58542306831580347587030344136, −16.06781569500563465953402002296, −14.53553600394254757875568809823, −13.58473867974590899241863940226, −13.22647844370310390354532295914, −10.641343700594707473916020624856, −9.2558695573597922692515198183, −8.97283598880206116520764182343, −7.27692775836328209668074364216, −6.33393903740371489427062475277, −4.751379778348255063826232293747, −3.19392386489661366793618584608, −1.10698758404243245275019233545, 1.63358996489598996843178257955, 2.67805953636830819358146684288, 3.85467952967504005813715253833, 5.77563652042077724942802997262, 7.5635448292360776139089328271, 9.029992793909947598518998712599, 9.63897572030548299029768045010, 10.68396634034751229768782927096, 12.62046527389620790259492605796, 13.059356580527553359432602368212, 14.30503907427709895316283596219, 15.4466691353949306889131616044, 17.22482462357902887013154673146, 18.3129968434968762620815002352, 19.061173032792603524921266421052, 20.23652714825067635420144032332, 20.98433067663258269334960594997, 22.02184758035111454130519096983, 22.91640274702844549669299663065, 24.89381140205483514187741073739, 25.66706262497286727057739024356, 26.3049762459088049673300653374, 27.693971611305098448429116862335, 28.67110927578741301356621854730, 29.65205000565725117787888085612

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