Properties

Degree 1
Conductor 83
Sign $0.742 + 0.670i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.665 + 0.746i)2-s + (−0.543 + 0.839i)3-s + (−0.114 − 0.993i)4-s + (0.720 − 0.693i)5-s + (−0.264 − 0.964i)6-s + (0.896 − 0.443i)7-s + (0.817 + 0.575i)8-s + (−0.409 − 0.912i)9-s + (0.0383 + 0.999i)10-s + (0.190 − 0.981i)11-s + (0.896 + 0.443i)12-s + (0.477 + 0.878i)13-s + (−0.264 + 0.964i)14-s + (0.190 + 0.981i)15-s + (−0.973 + 0.227i)16-s + (−0.997 + 0.0765i)17-s + ⋯
L(s,χ)  = 1  + (−0.665 + 0.746i)2-s + (−0.543 + 0.839i)3-s + (−0.114 − 0.993i)4-s + (0.720 − 0.693i)5-s + (−0.264 − 0.964i)6-s + (0.896 − 0.443i)7-s + (0.817 + 0.575i)8-s + (−0.409 − 0.912i)9-s + (0.0383 + 0.999i)10-s + (0.190 − 0.981i)11-s + (0.896 + 0.443i)12-s + (0.477 + 0.878i)13-s + (−0.264 + 0.964i)14-s + (0.190 + 0.981i)15-s + (−0.973 + 0.227i)16-s + (−0.997 + 0.0765i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.742 + 0.670i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.742 + 0.670i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(83\)
\( \varepsilon \)  =  $0.742 + 0.670i$
motivic weight  =  \(0\)
character  :  $\chi_{83} (40, \cdot )$
Sato-Tate  :  $\mu(41)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 83,\ (0:\ ),\ 0.742 + 0.670i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6673910357 + 0.2566513473i$
$L(\frac12,\chi)$  $\approx$  $0.6673910357 + 0.2566513473i$
$L(\chi,1)$  $\approx$  0.7329783633 + 0.2488519209i
$L(1,\chi)$  $\approx$  0.7329783633 + 0.2488519209i

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

−30.47751188007216593009376496280, −29.76513690823174296031597589117, −28.5650352972724519520366442513, −27.98055708318998944519060954593, −26.631066035624637440273242570891, −25.376109091417055797136971610306, −24.69754225252032942916210865894, −22.93793280539652966973763078494, −22.14156076098164441538187383381, −20.984324481104305189901285623880, −19.75592450621442655144545577621, −18.50504509756328912631445879161, −17.71426661207792235746722767996, −17.37871573644577583089841814470, −15.34401608110144285059648220545, −13.69866140978908857674234531685, −12.726469190135928095323045369571, −11.41453507469144002004630440615, −10.7315553825123875128794454366, −9.22055609656304912237461364661, −7.79381384027718821599733514614, −6.6890246532680693607530785574, −5.00939723422504066555454472201, −2.6595592482315232926011879732, −1.586837187824997882876345420398, 1.33403682442014400372287433456, 4.34889166129586111728274103266, 5.4291344103581620662027070790, 6.52644092782139932293438403941, 8.44264857685545706176972746873, 9.23579954797484719524222967496, 10.538971527449402097997496901067, 11.46733087113469376183671837220, 13.61738422648723441030233396847, 14.58112272869170148606250758412, 15.990275320848712665235509662492, 16.82432685638755149779219493997, 17.46740156956440357038593321947, 18.70057572312741475516370755794, 20.409358315970117691554893351441, 21.16427840126755802070233425449, 22.49472413966809085061369336204, 23.91266621799351324197884843400, 24.446922877541217343560751197635, 25.90351035532011882256428296231, 26.87863804884221759487914577442, 27.604268500655831158457530115974, 28.72481893405441406695161888798, 29.35122062114101479855012665349, 31.36977616269933679079652192154

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