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Properties

Label 1-1881-1881.151-r0-0-0
Degree $1$
Conductor $1881$
Sign $0.743 + 0.668i$
Analytic cond. $8.73532$
Root an. cond. $8.73532$
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  + (−0.978 + 0.207i)2-s + (0.913 − 0.406i)4-s + (−0.978 − 0.207i)5-s + (0.104 − 0.994i)7-s + (−0.809 + 0.587i)8-s + 10-s + (0.669 + 0.743i)13-s + (0.104 + 0.994i)14-s + (0.669 − 0.743i)16-s + (−0.309 − 0.951i)17-s + (−0.978 + 0.207i)20-s + (−0.5 + 0.866i)23-s + (0.913 + 0.406i)25-s + (−0.809 − 0.587i)26-s + (−0.309 − 0.951i)28-s + (−0.104 + 0.994i)29-s + ⋯
L(s)  = 1  + (−0.978 + 0.207i)2-s + (0.913 − 0.406i)4-s + (−0.978 − 0.207i)5-s + (0.104 − 0.994i)7-s + (−0.809 + 0.587i)8-s + 10-s + (0.669 + 0.743i)13-s + (0.104 + 0.994i)14-s + (0.669 − 0.743i)16-s + (−0.309 − 0.951i)17-s + (−0.978 + 0.207i)20-s + (−0.5 + 0.866i)23-s + (0.913 + 0.406i)25-s + (−0.809 − 0.587i)26-s + (−0.309 − 0.951i)28-s + (−0.104 + 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1881\)    =    \(3^{2} \cdot 11 \cdot 19\)
Sign: $0.743 + 0.668i$
Analytic conductor: \(8.73532\)
Root analytic conductor: \(8.73532\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1881} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1881,\ (0:\ ),\ 0.743 + 0.668i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6031418830 + 0.2314277288i\)
\(L(\frac12)\) \(\approx\) \(0.6031418830 + 0.2314277288i\)
\(L(1)\) \(\approx\) \(0.5891933697 + 0.01251408700i\)
\(L(1)\) \(\approx\) \(0.5891933697 + 0.01251408700i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.978 + 0.207i)T \)
5 \( 1 + (-0.978 - 0.207i)T \)
7 \( 1 + (0.104 - 0.994i)T \)
13 \( 1 + (0.669 + 0.743i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (-0.669 + 0.743i)T \)
89 \( 1 - T \)
97 \( 1 + (0.978 - 0.207i)T \)
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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

−19.96785529305079750630467021058, −19.12837209134630145261261455562, −18.57992850575330209156582655760, −18.03236123653815535111347780585, −17.20816587304655325438816417030, −16.26760961016663166243034915257, −15.6642685231027686688969012387, −15.179856942510610851407858296353, −14.428596700193307719156943250775, −12.91724924396464882023830067008, −12.461721133944791503152987910902, −11.65514363779361347089800915603, −11.0004165513606727451739835146, −10.395802236871914984602735278452, −9.38793754976910362392878578088, −8.38447665410179085221596748182, −8.30414972412914633503104703209, −7.311991257367741753072395603, −6.34308653583689261334513213798, −5.70721795430099107403154446262, −4.32499407978035135780697608465, −3.454950934656380753704164749397, −2.63871576407259069923834459254, −1.73295225408908040809858254115, −0.437216644944083831054977934865, 0.81096812175638217862051508276, 1.62359826778010175882108338530, 2.933287422212005105543141513077, 3.87223711878154261716059758913, 4.65838036085065324964640422829, 5.81671127914363202914329923253, 6.81128580399602319093433983032, 7.44200349212426186217352211392, 7.92591055715122809326347422916, 8.96750961831030357723785814010, 9.43821740585905296424400249929, 10.55693868024436907622497661855, 11.17044886817609222410611508781, 11.649751890208018377950405982319, 12.59873652070147728295871718652, 13.70900601098699504723456990608, 14.3435328192516196166243992323, 15.37930187339312764981200123886, 15.90421973841729015185845658300, 16.61199075353980912437486773574, 17.06435114707108814831844547648, 18.1764363657026757344631219981, 18.574866585892123701103140979, 19.52338180786773039738984966759, 20.0521856685388049866853537138

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