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Properties

Label 1-889-889.467-r0-0-0
Degree $1$
Conductor $889$
Sign $0.995 + 0.0915i$
Analytic cond. $4.12849$
Root an. cond. $4.12849$
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  + (−0.733 + 0.680i)2-s + (0.980 − 0.198i)3-s + (0.0747 − 0.997i)4-s + (0.826 − 0.563i)5-s + (−0.583 + 0.811i)6-s + (0.623 + 0.781i)8-s + (0.921 − 0.388i)9-s + (−0.222 + 0.974i)10-s + (−0.998 − 0.0498i)11-s + (−0.124 − 0.992i)12-s + (0.411 + 0.911i)13-s + (0.698 − 0.715i)15-s + (−0.988 − 0.149i)16-s + (0.969 − 0.246i)17-s + (−0.411 + 0.911i)18-s − 19-s + ⋯
L(s)  = 1  + (−0.733 + 0.680i)2-s + (0.980 − 0.198i)3-s + (0.0747 − 0.997i)4-s + (0.826 − 0.563i)5-s + (−0.583 + 0.811i)6-s + (0.623 + 0.781i)8-s + (0.921 − 0.388i)9-s + (−0.222 + 0.974i)10-s + (−0.998 − 0.0498i)11-s + (−0.124 − 0.992i)12-s + (0.411 + 0.911i)13-s + (0.698 − 0.715i)15-s + (−0.988 − 0.149i)16-s + (0.969 − 0.246i)17-s + (−0.411 + 0.911i)18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 889 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 889 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(889\)    =    \(7 \cdot 127\)
Sign: $0.995 + 0.0915i$
Analytic conductor: \(4.12849\)
Root analytic conductor: \(4.12849\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{889} (467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 889,\ (0:\ ),\ 0.995 + 0.0915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.756288585 + 0.08058212930i\)
\(L(\frac12)\) \(\approx\) \(1.756288585 + 0.08058212930i\)
\(L(1)\) \(\approx\) \(1.228996597 + 0.1130195765i\)
\(L(1)\) \(\approx\) \(1.228996597 + 0.1130195765i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (-0.733 + 0.680i)T \)
3 \( 1 + (0.980 - 0.198i)T \)
5 \( 1 + (0.826 - 0.563i)T \)
11 \( 1 + (-0.998 - 0.0498i)T \)
13 \( 1 + (0.411 + 0.911i)T \)
17 \( 1 + (0.969 - 0.246i)T \)
19 \( 1 - T \)
23 \( 1 + (0.998 - 0.0498i)T \)
29 \( 1 + (0.0249 + 0.999i)T \)
31 \( 1 + (0.969 + 0.246i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (-0.270 + 0.962i)T \)
43 \( 1 + (-0.878 - 0.478i)T \)
47 \( 1 + (-0.0747 + 0.997i)T \)
53 \( 1 + (0.124 - 0.992i)T \)
59 \( 1 + (0.766 - 0.642i)T \)
61 \( 1 + (-0.365 - 0.930i)T \)
67 \( 1 + (0.661 + 0.749i)T \)
71 \( 1 + (0.542 + 0.840i)T \)
73 \( 1 + (-0.955 + 0.294i)T \)
79 \( 1 + (0.542 + 0.840i)T \)
83 \( 1 + (-0.797 - 0.603i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (-0.853 - 0.521i)T \)
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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

−21.43594744125657382842251805343, −21.0987848063595989232747755276, −20.53635487037702271558896499732, −19.490415831652053443048775411746, −18.782601192506786645990608910716, −18.31322398046651034005632640259, −17.36147773724752236561417561609, −16.593179425661975956410811620599, −15.37872321035240107825315702158, −14.94093748262873238527570264535, −13.511807153536974410713129731095, −13.32548791699387664141632796648, −12.35961890590431351508330015518, −11.0075477380640556458758267342, −10.24663491284698238836724260665, −9.96815253974686860272366714901, −8.86200053813085995353047075294, −8.11293054241973211700420154734, −7.43834853725683015378706646773, −6.29489352940339163373762062106, −5.00977635309121806271139221121, −3.73406750413213577997620816061, −2.85284959919657699205805471091, −2.34922724709824839865032117249, −1.177415458148264171206533427202, 1.07895915768435761130935705888, 1.93540460169059164753915013644, 2.91129071079721406111750356358, 4.47678677941298512542714249338, 5.3118183649195722552865401389, 6.39856622019054939498611621238, 7.133602064002694684683767995908, 8.26703452728046545586963693473, 8.62639792699451604533278672849, 9.596252286172466345515530857161, 10.10499889257267099648594727102, 11.16252938101000903157063303859, 12.655685263742094247963208764659, 13.28513971848041760460530555903, 14.16391737030022344904096685217, 14.68617811128606304192033538914, 15.73985011353454617872468727862, 16.39736563431387578929991070914, 17.18221248629246216663617772690, 18.17264428104274608162854615740, 18.71157562643380820174199163045, 19.40407512120380972782279758850, 20.40957303378946859117893664008, 21.0307220853841286934972922380, 21.600501999412382384581491803157

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