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Properties

Label 1-83-83.78-r0-0-0
Degree $1$
Conductor $83$
Sign $-0.991 - 0.131i$
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  + (0.477 − 0.878i)2-s + (−0.264 − 0.964i)3-s + (−0.543 − 0.839i)4-s + (−0.771 + 0.636i)5-s + (−0.973 − 0.227i)6-s + (−0.665 − 0.746i)7-s + (−0.997 + 0.0765i)8-s + (−0.859 + 0.511i)9-s + (0.190 + 0.981i)10-s + (0.817 − 0.575i)11-s + (−0.665 + 0.746i)12-s + (0.606 − 0.795i)13-s + (−0.973 + 0.227i)14-s + (0.817 + 0.575i)15-s + (−0.409 + 0.912i)16-s + (−0.927 + 0.373i)17-s + ⋯
L(s)  = 1  + (0.477 − 0.878i)2-s + (−0.264 − 0.964i)3-s + (−0.543 − 0.839i)4-s + (−0.771 + 0.636i)5-s + (−0.973 − 0.227i)6-s + (−0.665 − 0.746i)7-s + (−0.997 + 0.0765i)8-s + (−0.859 + 0.511i)9-s + (0.190 + 0.981i)10-s + (0.817 − 0.575i)11-s + (−0.665 + 0.746i)12-s + (0.606 − 0.795i)13-s + (−0.973 + 0.227i)14-s + (0.817 + 0.575i)15-s + (−0.409 + 0.912i)16-s + (−0.927 + 0.373i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(83\)
Sign: $-0.991 - 0.131i$
Analytic conductor: \(0.385450\)
Root analytic conductor: \(0.385450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{83} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 83,\ (0:\ ),\ -0.991 - 0.131i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05218065242 - 0.7905776382i\)
\(L(\frac12)\) \(\approx\) \(0.05218065242 - 0.7905776382i\)
\(L(1)\) \(\approx\) \(0.5496656295 - 0.7235095732i\)
\(L(1)\) \(\approx\) \(0.5496656295 - 0.7235095732i\)

Euler product

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

−31.31627910457141149261718208336, −31.171689524395863445924802047708, −29.007860524926009247717788310744, −27.94404088736116588195230033414, −27.12547700974422107958813404674, −26.00488102913491256066678660633, −25.018432298325488997353779232, −23.807131680817353843286211799890, −22.75149705034792571308536237128, −22.053013793509794789071508344479, −20.87154423013596017536839552712, −19.64420415891106456185444380196, −17.94524481925143987750908429803, −16.66806721130211144936694465684, −15.88109497274240867055448296638, −15.284468281628507499815297652017, −13.88459024725852851862490584727, −12.315893311403804686756918526690, −11.55782268038262499611970983485, −9.33042083177724841010798311817, −8.817547161371300351088891684422, −7.00391253568421978409093548518, −5.64267948340131258768671896026, −4.453525170522151707296270738730, −3.457954087571255053816876163914, 0.815236042323645389927492298540, 2.84246514862309621577873834592, 3.98833868750031452099225337312, 5.98576381108846083005965430241, 7.03517808962717517281606840274, 8.64711427042757425744173241379, 10.52340557319031042422876407128, 11.309599950215721890344321137, 12.434130208426622933102168016317, 13.48368430785477614493983758725, 14.4057382596461191680007147496, 15.974058909552629249071988880803, 17.63687227050735378908484626358, 18.70000994333034342166705577667, 19.62099945567106554926675163251, 20.20406999105459973221980505443, 22.22596310178353875622516273957, 22.725176695075658719030643369552, 23.65105681274879752057046251699, 24.66257547117342909971811242043, 26.317403499563308402102476685717, 27.38264193177434622904122654138, 28.61847969144331011893273800864, 29.53729545314569268195402922331, 30.40514451611457770568614286608

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