Properties

Degree 1
Conductor $ 3 \cdot 23 $
Sign $0.764 - 0.644i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.959 − 0.281i)2-s + (0.841 − 0.540i)4-s + (−0.142 − 0.989i)5-s + (−0.415 + 0.909i)7-s + (0.654 − 0.755i)8-s + (−0.415 − 0.909i)10-s + (−0.959 − 0.281i)11-s + (0.415 + 0.909i)13-s + (−0.142 + 0.989i)14-s + (0.415 − 0.909i)16-s + (0.841 + 0.540i)17-s + (−0.841 + 0.540i)19-s + (−0.654 − 0.755i)20-s − 22-s + (−0.959 + 0.281i)25-s + (0.654 + 0.755i)26-s + ⋯
L(s,χ)  = 1  + (0.959 − 0.281i)2-s + (0.841 − 0.540i)4-s + (−0.142 − 0.989i)5-s + (−0.415 + 0.909i)7-s + (0.654 − 0.755i)8-s + (−0.415 − 0.909i)10-s + (−0.959 − 0.281i)11-s + (0.415 + 0.909i)13-s + (−0.142 + 0.989i)14-s + (0.415 − 0.909i)16-s + (0.841 + 0.540i)17-s + (−0.841 + 0.540i)19-s + (−0.654 − 0.755i)20-s − 22-s + (−0.959 + 0.281i)25-s + (0.654 + 0.755i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.764 - 0.644i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 69 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.764 - 0.644i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(69\)    =    \(3 \cdot 23\)
\( \varepsilon \)  =  $0.764 - 0.644i$
motivic weight  =  \(0\)
character  :  $\chi_{69} (20, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 69,\ (0:\ ),\ 0.764 - 0.644i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.377938426 - 0.5032153484i$
$L(\frac12,\chi)$  $\approx$  $1.377938426 - 0.5032153484i$
$L(\chi,1)$  $\approx$  1.497923106 - 0.3812671157i
$L(1,\chi)$  $\approx$  1.497923106 - 0.3812671157i

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

−32.01812046447165901881754587151, −30.86690972492260914619378385967, −29.95002660786973928454729967565, −29.27293076044455545941056800932, −27.48187362104359267544923620962, −25.995485362609902668786867452330, −25.688036726869434397901435797633, −23.905098924510290528678325377300, −23.086939674898868441637212100091, −22.37116540974269782030749939326, −20.9884228055221157183924248658, −20.00469212885236328962554433005, −18.538916294070923517380226614097, −17.1210036629957997336706135248, −15.81597843994239708088872520234, −14.88116869887037289439033668173, −13.678640545564170070478381393188, −12.74437814802936570016273337101, −11.10284672644128410251800238965, −10.26655540541446648592102953668, −7.80547780273660577088821574584, −6.94654265525579188393662983138, −5.54435445472554231933230711262, −3.85535792105064962780380100994, −2.71256253902432592314717151711, 1.95354316575904223609827617475, 3.68429268297079273548229110118, 5.15760963915594365849335081010, 6.16601445759224821934210875586, 8.11143060697423245535834361267, 9.59066027977716646664127399093, 11.18207390114359311505688491320, 12.44748637509047669972940995232, 13.06437915589755892269146201137, 14.56358905329548413198528022595, 15.84034677666190876619865632287, 16.57158297290392375106986898013, 18.64955539727148244852119121247, 19.59203223771486213038273464965, 21.09950690773700674539283330027, 21.40609450997760720341155527507, 23.0528459169545313067407587704, 23.86788724597431353381957395310, 24.90209294308098071488996444712, 25.93791506699985795316959960291, 27.91962018142597418651897304930, 28.57243161795002295421744670136, 29.50661327860750454316731493317, 31.01278158255213694336585770935, 31.73391869666421496660240286791

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