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Properties

Label 1-4563-4563.1064-r0-0-0
Degree $1$
Conductor $4563$
Sign $0.390 + 0.920i$
Analytic cond. $21.1904$
Root an. cond. $21.1904$
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  + (0.556 − 0.830i)2-s + (−0.379 − 0.925i)4-s + (−0.265 − 0.964i)5-s + (−0.852 − 0.523i)7-s + (−0.979 − 0.200i)8-s + (−0.948 − 0.316i)10-s + (0.914 − 0.404i)11-s + (−0.909 + 0.416i)14-s + (−0.711 + 0.702i)16-s + (−0.748 − 0.663i)17-s + (−0.866 − 0.5i)19-s + (−0.791 + 0.611i)20-s + (0.173 − 0.984i)22-s + (0.173 − 0.984i)23-s + (−0.859 + 0.511i)25-s + ⋯
L(s)  = 1  + (0.556 − 0.830i)2-s + (−0.379 − 0.925i)4-s + (−0.265 − 0.964i)5-s + (−0.852 − 0.523i)7-s + (−0.979 − 0.200i)8-s + (−0.948 − 0.316i)10-s + (0.914 − 0.404i)11-s + (−0.909 + 0.416i)14-s + (−0.711 + 0.702i)16-s + (−0.748 − 0.663i)17-s + (−0.866 − 0.5i)19-s + (−0.791 + 0.611i)20-s + (0.173 − 0.984i)22-s + (0.173 − 0.984i)23-s + (−0.859 + 0.511i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $0.390 + 0.920i$
Analytic conductor: \(21.1904\)
Root analytic conductor: \(21.1904\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4563} (1064, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4563,\ (0:\ ),\ 0.390 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.8179287577 - 0.5415979700i\)
\(L(\frac12)\) \(\approx\) \(-0.8179287577 - 0.5415979700i\)
\(L(1)\) \(\approx\) \(0.5292745985 - 0.8472978839i\)
\(L(1)\) \(\approx\) \(0.5292745985 - 0.8472978839i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.556 - 0.830i)T \)
5 \( 1 + (-0.265 - 0.964i)T \)
7 \( 1 + (-0.852 - 0.523i)T \)
11 \( 1 + (0.914 - 0.404i)T \)
17 \( 1 + (-0.748 - 0.663i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.830 - 0.556i)T \)
31 \( 1 + (0.999 + 0.0134i)T \)
37 \( 1 + (-0.160 - 0.987i)T \)
41 \( 1 + (-0.682 - 0.730i)T \)
43 \( 1 + (-0.0134 - 0.999i)T \)
47 \( 1 + (-0.133 - 0.991i)T \)
53 \( 1 + (0.748 + 0.663i)T \)
59 \( 1 + (0.702 - 0.711i)T \)
61 \( 1 + (-0.783 - 0.621i)T \)
67 \( 1 + (-0.213 + 0.977i)T \)
71 \( 1 + (-0.992 - 0.120i)T \)
73 \( 1 + (0.721 - 0.692i)T \)
79 \( 1 + (0.611 + 0.791i)T \)
83 \( 1 + (0.291 + 0.956i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.967 - 0.252i)T \)
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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

−18.71445639511986351043511965027, −17.97571112032702958399122148029, −17.33059211853805520088635880854, −16.69585813135971946442353065866, −15.943374441382276291380038485587, −15.14286874113295641211480078773, −15.0013386853275290682224222043, −14.27418193807491474657074759958, −13.32131483368128555380678337739, −12.96979222976746632889462990143, −11.94363638728019187393938489452, −11.667233980974585861258766755955, −10.61343247397227003267252163894, −9.77925357973921622573251546478, −9.09865446538184471255115363654, −8.35916709288611787839295416492, −7.57421868735317291499064821320, −6.75071454063593664437201731665, −6.38589933894819148443536967270, −5.85152863729535581794490544859, −4.72591628509595334172980198948, −3.973662527093399539466078895224, −3.37274205346196222966392910881, −2.67678750230394296205384132409, −1.676246160300513796724891544297, 0.2752424816565646727716535152, 0.78982968174390600919577954395, 1.91869614805185649229810451256, 2.67209410741592688729447881837, 3.78586820266743656967980199525, 4.05459446905027603839711737035, 4.841828690201150149374750285320, 5.654209374109853255696150515996, 6.51221439041775277214239912028, 7.01404101354247583350783983308, 8.339659411743411164323194569101, 9.01136746598364547845949112699, 9.39128063283968529064238016803, 10.30999988158102054275206915203, 10.947725916620828551058741072333, 11.729511227979677253093819752837, 12.282014923844433689165877208721, 12.925402422803904479977449196648, 13.57526022365560726284738178677, 13.930269499930161156779488997509, 15.0303282252503503898679906424, 15.544992924721697718724263123809, 16.38695861419183838242389359531, 16.957739010929215971145259693223, 17.61859737602391447320757289284

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