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Properties

Label 1-83-83.27-r0-0-0
Degree $1$
Conductor $83$
Sign $0.742 - 0.670i$
Analytic cond. $0.385450$
Root an. cond. $0.385450$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $0.742 - 0.670i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ 0.742 - 0.670i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6673910357 - 0.2566513473i\)
\(L(\frac12)\) \(\approx\) \(0.6673910357 - 0.2566513473i\)
\(L(1)\) \(\approx\) \(0.7329783633 - 0.2488519209i\)
\(L(1)\) \(\approx\) \(0.7329783633 - 0.2488519209i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad83 \( 1 \)
good2 \( 1 + (-0.665 - 0.746i)T \)
3 \( 1 + (-0.543 - 0.839i)T \)
5 \( 1 + (0.720 + 0.693i)T \)
7 \( 1 + (0.896 + 0.443i)T \)
11 \( 1 + (0.190 + 0.981i)T \)
13 \( 1 + (0.477 - 0.878i)T \)
17 \( 1 + (-0.997 - 0.0765i)T \)
19 \( 1 + (0.338 - 0.941i)T \)
23 \( 1 + (0.953 + 0.301i)T \)
29 \( 1 + (-0.771 - 0.636i)T \)
31 \( 1 + (0.817 + 0.575i)T \)
37 \( 1 + (-0.409 - 0.912i)T \)
41 \( 1 + (-0.665 + 0.746i)T \)
43 \( 1 + (0.988 + 0.152i)T \)
47 \( 1 + (-0.859 - 0.511i)T \)
53 \( 1 + (-0.859 + 0.511i)T \)
59 \( 1 + (-0.927 - 0.373i)T \)
61 \( 1 + (0.606 - 0.795i)T \)
67 \( 1 + (-0.973 - 0.227i)T \)
71 \( 1 + (0.896 - 0.443i)T \)
73 \( 1 + (0.0383 - 0.999i)T \)
79 \( 1 + (-0.114 + 0.993i)T \)
89 \( 1 + (-0.264 + 0.964i)T \)
97 \( 1 + (-0.264 - 0.964i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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