Properties

Degree 1
Conductor 23
Sign $-0.771 + 0.635i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.142 + 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (0.654 + 0.755i)5-s + (−0.959 + 0.281i)6-s + (−0.841 − 0.540i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.841 + 0.540i)10-s + (0.142 + 0.989i)11-s + (−0.142 − 0.989i)12-s + (0.841 − 0.540i)13-s + (0.654 − 0.755i)14-s + (−0.415 + 0.909i)15-s + (0.841 + 0.540i)16-s + (0.959 − 0.281i)17-s + ⋯
L(s,χ)  = 1  + (−0.142 + 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (0.654 + 0.755i)5-s + (−0.959 + 0.281i)6-s + (−0.841 − 0.540i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (−0.841 + 0.540i)10-s + (0.142 + 0.989i)11-s + (−0.142 − 0.989i)12-s + (0.841 − 0.540i)13-s + (0.654 − 0.755i)14-s + (−0.415 + 0.909i)15-s + (0.841 + 0.540i)16-s + (0.959 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.771 + 0.635i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.771 + 0.635i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(23\)
\( \varepsilon \)  =  $-0.771 + 0.635i$
motivic weight  =  \(0\)
character  :  $\chi_{23} (10, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 23,\ (1:\ ),\ -0.771 + 0.635i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4325295310 + 1.205465147i$
$L(\frac12,\chi)$  $\approx$  $0.4325295310 + 1.205465147i$
$L(\chi,1)$  $\approx$  0.7210110963 + 0.8129020667i
$L(1,\chi)$  $\approx$  0.7210110963 + 0.8129020667i

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

−37.945948040157155051117051391572, −37.005235254945245174840162107254, −35.90667704831351102638463650205, −35.01877570996695299175752364099, −32.456603073918669037352085639277, −31.62789800199476039996211751489, −30.201494835137184619792322258711, −29.08399409536481223018386105355, −28.32397460964055898121881676704, −26.28902531255279358704820131008, −25.08623284191377978574235623390, −23.55878638737965249371350251604, −21.834517033007257391640124424978, −20.56681531641732360817054936241, −19.23240608136633369695707634318, −18.28452522304774822448264638292, −16.63833488600076627074233743463, −13.906373355556694967141597978376, −13.016062555098127733237425238800, −11.76456311027407169240600818462, −9.51609515046732405008325293569, −8.455756935396261162415773663793, −5.88611725344099459803832661243, −3.147747137331669402331371101034, −1.23377715818156234577714895682, 3.60810710722134433432466230901, 5.67852031718441859245797855795, 7.41010668689748212895182763825, 9.45987916574414404551575673409, 10.29827245742697055683394829985, 13.43210767507056865197446793618, 14.590522827823499865063779504670, 15.8331749844610280629078898790, 17.132093479419417065521898332411, 18.675603964503428488534354330079, 20.46617546727517632449792563656, 22.287419380138063375714722575194, 22.98906973737863461487422124845, 25.34252927830726541670301819878, 25.85554676740181220977013019623, 27.0048933421035340969024919826, 28.39794589033058331560831436561, 30.38871356131503417731294508904, 31.95226824792651724429364040440, 33.09903504809178900454122103866, 33.62823361044807264731901948157, 35.29063164677407843707684534234, 36.6636214515049842641653433633, 37.787570713384329151713533241753, 39.07262620557626963054299755984

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