Properties

Degree 1
Conductor $ 2^{2} \cdot 23 $
Sign $0.529 + 0.848i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.654 + 0.755i)3-s + (0.142 + 0.989i)5-s + (0.415 − 0.909i)7-s + (−0.142 + 0.989i)9-s + (−0.959 − 0.281i)11-s + (0.415 + 0.909i)13-s + (−0.654 + 0.755i)15-s + (−0.841 − 0.540i)17-s + (0.841 − 0.540i)19-s + (0.959 − 0.281i)21-s + (−0.959 + 0.281i)25-s + (−0.841 + 0.540i)27-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.415 − 0.909i)33-s + ⋯
L(s,χ)  = 1  + (0.654 + 0.755i)3-s + (0.142 + 0.989i)5-s + (0.415 − 0.909i)7-s + (−0.142 + 0.989i)9-s + (−0.959 − 0.281i)11-s + (0.415 + 0.909i)13-s + (−0.654 + 0.755i)15-s + (−0.841 − 0.540i)17-s + (0.841 − 0.540i)19-s + (0.959 − 0.281i)21-s + (−0.959 + 0.281i)25-s + (−0.841 + 0.540i)27-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.415 − 0.909i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.529 + 0.848i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.529 + 0.848i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(92\)    =    \(2^{2} \cdot 23\)
\( \varepsilon \)  =  $0.529 + 0.848i$
motivic weight  =  \(0\)
character  :  $\chi_{92} (43, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 92,\ (0:\ ),\ 0.529 + 0.848i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.061752144 + 0.5892415556i$
$L(\frac12,\chi)$  $\approx$  $1.061752144 + 0.5892415556i$
$L(\chi,1)$  $\approx$  1.168571365 + 0.3988168022i
$L(1,\chi)$  $\approx$  1.168571365 + 0.3988168022i

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

−30.436184063525679307130013491475, −28.94695516352197558087512426759, −28.411579962794133900056682953346, −27.03474674818034429491694343586, −25.68706110858232847552923221009, −24.86912340038040220788841208447, −24.19363233549228004448566827660, −23.05085135889571442489497774631, −21.37242449735258548692452443965, −20.569023916213222789878824596237, −19.62772595501735741122619959323, −18.24484145179155861631994273982, −17.67051522115026834071537387289, −15.91084019527906837991193756389, −15.014813865132945693216720928572, −13.51313457659889814148933536091, −12.76653992671027723839694143988, −11.7579440558038620016128101593, −9.88293503915710432484369458360, −8.486367546889064038382085299126, −8.01573352111948690496932249241, −6.14547950618245809659829913400, −4.89342566782833053586186581796, −2.90408157574745849165442649621, −1.50984416072749612545460215344, 2.3708199900890198514253246559, 3.65630477719136189554145260427, 4.96103933626250544064299173520, 6.8419086715596014732595110305, 7.97963675357926205589800191303, 9.42107165116403880562799973391, 10.59338793601727773134610896048, 11.27311281570759548725257694492, 13.63218580681171477165114361676, 14.00018265883928437487019280554, 15.34095639827908844727841077923, 16.24193747203072613137506235514, 17.70643734673537649137229845962, 18.823508108386155323461409754879, 20.01940238039879443168631221083, 21.002249226927515790900569281395, 21.905363016018520410127183197232, 23.038590492736754945785161224431, 24.21288268971400388150175225823, 25.666363257889887911242065448741, 26.62332079461077617596027325204, 26.80805101014496597192742430060, 28.408647059486744645756805332373, 29.584521099411673939731538312342, 30.795106063783915902594235561211

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