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Properties

Label 1-1328-1328.147-r1-0-0
Degree $1$
Conductor $1328$
Sign $0.903 + 0.429i$
Analytic cond. $142.713$
Root an. cond. $142.713$
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  + (0.993 + 0.114i)3-s + (−0.152 − 0.988i)5-s + (−0.859 − 0.511i)7-s + (0.973 + 0.227i)9-s + (0.999 + 0.0383i)11-s + (0.746 − 0.665i)13-s + (−0.0383 − 0.999i)15-s + (0.817 + 0.575i)17-s + (0.373 + 0.927i)19-s + (−0.795 − 0.606i)21-s + (−0.771 + 0.636i)23-s + (−0.953 + 0.301i)25-s + (0.941 + 0.338i)27-s + (0.693 + 0.720i)29-s + (−0.190 + 0.981i)31-s + ⋯
L(s)  = 1  + (0.993 + 0.114i)3-s + (−0.152 − 0.988i)5-s + (−0.859 − 0.511i)7-s + (0.973 + 0.227i)9-s + (0.999 + 0.0383i)11-s + (0.746 − 0.665i)13-s + (−0.0383 − 0.999i)15-s + (0.817 + 0.575i)17-s + (0.373 + 0.927i)19-s + (−0.795 − 0.606i)21-s + (−0.771 + 0.636i)23-s + (−0.953 + 0.301i)25-s + (0.941 + 0.338i)27-s + (0.693 + 0.720i)29-s + (−0.190 + 0.981i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1328 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1328 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1328\)    =    \(2^{4} \cdot 83\)
Sign: $0.903 + 0.429i$
Analytic conductor: \(142.713\)
Root analytic conductor: \(142.713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1328} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1328,\ (1:\ ),\ 0.903 + 0.429i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.163473631 + 0.7133252854i\)
\(L(\frac12)\) \(\approx\) \(3.163473631 + 0.7133252854i\)
\(L(1)\) \(\approx\) \(1.546595379 - 0.05180730386i\)
\(L(1)\) \(\approx\) \(1.546595379 - 0.05180730386i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
83 \( 1 \)
good3 \( 1 + (0.993 + 0.114i)T \)
5 \( 1 + (-0.152 - 0.988i)T \)
7 \( 1 + (-0.859 - 0.511i)T \)
11 \( 1 + (0.999 + 0.0383i)T \)
13 \( 1 + (0.746 - 0.665i)T \)
17 \( 1 + (0.817 + 0.575i)T \)
19 \( 1 + (0.373 + 0.927i)T \)
23 \( 1 + (-0.771 + 0.636i)T \)
29 \( 1 + (0.693 + 0.720i)T \)
31 \( 1 + (-0.190 + 0.981i)T \)
37 \( 1 + (0.227 + 0.973i)T \)
41 \( 1 + (-0.896 + 0.443i)T \)
43 \( 1 + (-0.941 + 0.338i)T \)
47 \( 1 + (0.409 + 0.912i)T \)
53 \( 1 + (0.912 + 0.409i)T \)
59 \( 1 + (-0.0765 - 0.997i)T \)
61 \( 1 + (0.878 + 0.477i)T \)
67 \( 1 + (0.964 + 0.264i)T \)
71 \( 1 + (-0.859 + 0.511i)T \)
73 \( 1 + (-0.953 - 0.301i)T \)
79 \( 1 + (-0.606 - 0.795i)T \)
89 \( 1 + (0.543 - 0.839i)T \)
97 \( 1 + (-0.543 - 0.839i)T \)
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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

−20.553850488841937043849371454049, −19.76814540710566600401935809475, −19.165731569154361864767972710901, −18.58468122556071746758907332444, −18.02992205588142741616722641281, −16.70777843376774097667423865961, −15.92216217590194414455062670029, −15.27495470083411675410774173079, −14.50580096036697694900538424757, −13.84794568457385826098671022082, −13.26798879208616363472351979616, −12.02986951762867748275403808574, −11.62075448275530640309552234465, −10.33477144869158410392623003929, −9.66109586064161685866247795775, −8.98955510023455760373734785939, −8.16979126104514089102884380020, −7.031346484753230382198414457696, −6.68735314632262447133611333285, −5.72704457697889883531051502562, −4.13536927786046133573878439346, −3.595951176405859852800468518437, −2.72870654009634086870205422279, −1.99831863667797626602869919179, −0.5882199668023975899048215956, 1.10929049518170415697772878473, 1.515914273040376157713500544484, 3.24672371847110191763905580012, 3.59679989524493240163889370198, 4.45969866539653894906878551810, 5.62727591849664850443104512776, 6.548363010037617288377440998398, 7.58702583683224944938996102722, 8.33099306627959851538933682831, 8.92531745847680150569722898403, 9.938303383476201836268561288403, 10.21712145479988638665546845704, 11.72327185106744446078957709671, 12.50335734271932156958036471957, 13.121242264388622622497653255027, 13.87639065003524484959033716755, 14.548316762164742510109894216770, 15.58058056301677200279335832994, 16.19969474757019952890736613026, 16.74602992450724586869876357657, 17.7287150805511826925351524525, 18.79915623940487248543260424953, 19.464424334983574715303186202004, 20.2453012654452804621375954046, 20.35676103623599446384591598151

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