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Properties

Label 1-319-319.128-r0-0-0
Degree $1$
Conductor $319$
Sign $0.421 - 0.906i$
Analytic cond. $1.48142$
Root an. cond. $1.48142$
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  + (0.951 − 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.309 − 0.951i)6-s + (0.809 − 0.587i)7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + i·10-s − i·12-s + (0.309 + 0.951i)13-s + (0.587 − 0.809i)14-s + (0.587 + 0.809i)15-s + (0.309 − 0.951i)16-s + (−0.951 − 0.309i)17-s + (−0.587 − 0.809i)18-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (0.309 − 0.951i)6-s + (0.809 − 0.587i)7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + i·10-s − i·12-s + (0.309 + 0.951i)13-s + (0.587 − 0.809i)14-s + (0.587 + 0.809i)15-s + (0.309 − 0.951i)16-s + (−0.951 − 0.309i)17-s + (−0.587 − 0.809i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(319\)    =    \(11 \cdot 29\)
Sign: $0.421 - 0.906i$
Analytic conductor: \(1.48142\)
Root analytic conductor: \(1.48142\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{319} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 319,\ (0:\ ),\ 0.421 - 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.298431607 - 1.466611677i\)
\(L(\frac12)\) \(\approx\) \(2.298431607 - 1.466611677i\)
\(L(1)\) \(\approx\) \(1.969713667 - 0.8158712504i\)
\(L(1)\) \(\approx\) \(1.969713667 - 0.8158712504i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (-0.587 + 0.809i)T \)
23 \( 1 + T \)
31 \( 1 + (-0.951 + 0.309i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.587 - 0.809i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (-0.951 - 0.309i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (0.951 - 0.309i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.951 - 0.309i)T \)
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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

−25.00237898556459419998129966055, −24.58901508829854979369877002502, −23.58426963450734624364356767399, −22.57747145138544992440696822034, −21.520687990560396870905131242158, −21.1226602864384075809816680625, −20.15191338529568606403036078333, −19.62615022030843978781150000903, −17.834685673446630973194356023147, −16.861148676672509600677495695405, −15.94097377330563390137560607762, −15.17636481883180801406917092836, −14.71198631380233332620819729679, −13.30107900393986176794478476096, −12.8340265758598796186755955286, −11.429788928905716553275286191677, −10.8381435332592640567699379587, −9.11667428577254030258629567808, −8.425506515595833227528960657152, −7.58473074442533043872991905223, −5.88228384298036932340527115346, −4.915529016063408451616918498513, −4.3681585376316529177396228753, −3.11833956380532663456438694602, −1.94281547192344600712974598763, 1.49334617639516090936940977479, 2.43878514299481468885099256884, 3.596803797426690241409077111186, 4.48350955971953564978303327662, 6.12502237622174983141798980601, 6.96805168158769854964504458124, 7.645497047813495114145122233518, 9.02690377279753231734951527197, 10.58178451733003498160650592083, 11.27066081075708179731598424245, 12.125862466198467121416519670789, 13.30512625633963677192836579118, 14.03486857014266607311350589965, 14.642604688912718148787987242, 15.4208154398111197013500025217, 16.83720585381644720730244905728, 18.13720619515626823664499352519, 18.89248090630589851780717906545, 19.68032552484494361339064230370, 20.58701630701660332454490232023, 21.32919346542754795251309057233, 22.46173104709271562072890532479, 23.429493994820739337646284176551, 23.81497634419343336069901809679, 24.79525161367746171471168023551

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