Properties

Degree 1
Conductor $ 3^{2} \cdot 11 $
Sign $0.296 + 0.954i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.104 − 0.994i)5-s + (−0.669 + 0.743i)7-s + (−0.309 + 0.951i)8-s − 10-s + (−0.913 + 0.406i)13-s + (0.669 + 0.743i)14-s + (0.913 + 0.406i)16-s + (0.809 − 0.587i)17-s + (−0.309 + 0.951i)19-s + (−0.104 + 0.994i)20-s + (−0.5 + 0.866i)23-s + (−0.978 + 0.207i)25-s + (0.309 + 0.951i)26-s + ⋯
L(s,χ)  = 1  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s + (−0.104 − 0.994i)5-s + (−0.669 + 0.743i)7-s + (−0.309 + 0.951i)8-s − 10-s + (−0.913 + 0.406i)13-s + (0.669 + 0.743i)14-s + (0.913 + 0.406i)16-s + (0.809 − 0.587i)17-s + (−0.309 + 0.951i)19-s + (−0.104 + 0.994i)20-s + (−0.5 + 0.866i)23-s + (−0.978 + 0.207i)25-s + (0.309 + 0.951i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.296 + 0.954i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 99 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.296 + 0.954i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(99\)    =    \(3^{2} \cdot 11\)
\( \varepsilon \)  =  $0.296 + 0.954i$
motivic weight  =  \(0\)
character  :  $\chi_{99} (79, \cdot )$
Sato-Tate  :  $\mu(30)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 99,\ (1:\ ),\ 0.296 + 0.954i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.2355640385 + 0.1734834668i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.2355640385 + 0.1734834668i\)
\(L(\chi,1)\)  \(\approx\)  \(0.6355331943 - 0.3092955316i\)
\(L(1,\chi)\)  \(\approx\)  \(0.6355331943 - 0.3092955316i\)

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

−29.8977707547454838151355415418, −28.31116140607085728768639346696, −26.99273952739150634378040429949, −26.3463427746907266156547200384, −25.56518380931081953553094689505, −24.30921160163818753115609179432, −23.206869151753660604101751004944, −22.53276375903058496712677032317, −21.56230769118084972588489930382, −19.73063379658382536917765031085, −18.82244126724806026825487824128, −17.59658962972232127619950764196, −16.719465781851607219031683320717, −15.46519769506742951794830479351, −14.58187887277735151864538450877, −13.57842135249378029466269470603, −12.37734153949050987282679188431, −10.5526437448217451589718163304, −9.643818659468679680899494045651, −7.938286135399354118291044914126, −7.01177604325322829435942043928, −6.029285686475049446157826073458, −4.352001594175097078929798016196, −3.03937242581818596294809746711, −0.124522044588925134355987689496, 1.66910827511000909401424042793, 3.22157225253061690379801663636, 4.68969484904433095447883344021, 5.80052629114344919030964273590, 7.97781961250572526437419842861, 9.26482246188996720005356017838, 9.930406737413477562449580976779, 11.76974983413793651593517467035, 12.32649258377635330403695074267, 13.37195180676148847565805548864, 14.68934068121204344821114450096, 16.17011113278204225588283560068, 17.25017417849306268379846627546, 18.67089629528942727319938131274, 19.44725085005870626549239518069, 20.51369002720823583041741595855, 21.4386451598623258885293821736, 22.41682475112391984162613502764, 23.533333640254929264526345379987, 24.66078186882191030493802826993, 25.86065462960693485711471847185, 27.33967804175928394791682959528, 27.96288184739649309621006117998, 29.071240130104564537786693154494, 29.5836491883234055440289324471

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