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Properties

Label 1-319-319.226-r1-0-0
Degree $1$
Conductor $319$
Sign $-0.285 - 0.958i$
Analytic cond. $34.2813$
Root an. cond. $34.2813$
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  + (−0.753 − 0.657i)2-s + (0.134 − 0.990i)3-s + (0.134 + 0.990i)4-s + (−0.393 + 0.919i)5-s + (−0.753 + 0.657i)6-s + (0.691 − 0.722i)7-s + (0.550 − 0.834i)8-s + (−0.963 − 0.266i)9-s + (0.900 − 0.433i)10-s + 12-s + (0.0448 − 0.998i)13-s + (−0.995 + 0.0896i)14-s + (0.858 + 0.512i)15-s + (−0.963 + 0.266i)16-s + (0.809 + 0.587i)17-s + (0.550 + 0.834i)18-s + ⋯
L(s)  = 1  + (−0.753 − 0.657i)2-s + (0.134 − 0.990i)3-s + (0.134 + 0.990i)4-s + (−0.393 + 0.919i)5-s + (−0.753 + 0.657i)6-s + (0.691 − 0.722i)7-s + (0.550 − 0.834i)8-s + (−0.963 − 0.266i)9-s + (0.900 − 0.433i)10-s + 12-s + (0.0448 − 0.998i)13-s + (−0.995 + 0.0896i)14-s + (0.858 + 0.512i)15-s + (−0.963 + 0.266i)16-s + (0.809 + 0.587i)17-s + (0.550 + 0.834i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(319\)    =    \(11 \cdot 29\)
Sign: $-0.285 - 0.958i$
Analytic conductor: \(34.2813\)
Root analytic conductor: \(34.2813\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{319} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 319,\ (1:\ ),\ -0.285 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7808055129 - 1.047577238i\)
\(L(\frac12)\) \(\approx\) \(0.7808055129 - 1.047577238i\)
\(L(1)\) \(\approx\) \(0.7126061889 - 0.4254585153i\)
\(L(1)\) \(\approx\) \(0.7126061889 - 0.4254585153i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.753 - 0.657i)T \)
3 \( 1 + (0.134 - 0.990i)T \)
5 \( 1 + (-0.393 + 0.919i)T \)
7 \( 1 + (0.691 - 0.722i)T \)
13 \( 1 + (0.0448 - 0.998i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (0.691 + 0.722i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (0.753 + 0.657i)T \)
37 \( 1 + (0.936 - 0.351i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (0.936 + 0.351i)T \)
53 \( 1 + (0.753 + 0.657i)T \)
59 \( 1 + (0.309 - 0.951i)T \)
61 \( 1 + (-0.473 - 0.880i)T \)
67 \( 1 + (0.623 + 0.781i)T \)
71 \( 1 + (-0.963 + 0.266i)T \)
73 \( 1 + (-0.858 - 0.512i)T \)
79 \( 1 + (0.963 + 0.266i)T \)
83 \( 1 + (-0.983 - 0.178i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (0.473 - 0.880i)T \)
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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

−25.19083239603087323064681282799, −24.478817319134056964709547139113, −23.68345196595970082513476884751, −22.62331421021131846291505445311, −21.40415854271471260611991851516, −20.66491226677822689519445344880, −19.87093341347340481860547020881, −18.84415065268353937923708949404, −17.863638870838617684731293260188, −16.69473826446705207215154812074, −16.32657398562973954706256499566, −15.37681371115101570843485848934, −14.64521776476250946707292015079, −13.67310580559823180962530897122, −11.849341141297377877873471568734, −11.3482789905891870954346575299, −9.942911835507119064338207789076, −9.16375391555208521039879373073, −8.50620254046276473933059498092, −7.62091125143771760486916282430, −6.027671264131208481811945813294, −5.0163056380491732050287308790, −4.40286439830944039882270776307, −2.51460156403157096584480824048, −0.949745589581379588668620942797, 0.650174316098318877372499513205, 1.698737527809078851715210944544, 2.98750335360999522215647347783, 3.8090074083564682560546623840, 5.77752232428223107924858157081, 7.21196846160991300383546162193, 7.67510444010936621875707585023, 8.42214758901098143744035029928, 10.011863606550015893045757809315, 10.7674448969482915099474009, 11.6831270438879119012641187916, 12.43609493322299366773704808068, 13.624111304768036258082211114166, 14.393516688888980098489163689768, 15.62914317300535593636218095382, 17.051843022450937842187279743348, 17.69705168995880148479052576976, 18.440162493409890415277792392699, 19.20715657290529257249738738859, 20.01159634102255636616191339539, 20.73216775712928688035785003275, 21.95904819915635022729682043151, 22.990131129327854612189178029501, 23.649120318951640887675036791631, 24.93616355279682023639640929963

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