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Properties

Label 1-147-147.134-r1-0-0
Degree $1$
Conductor $147$
Sign $-0.462 - 0.886i$
Analytic cond. $15.7973$
Root an. cond. $15.7973$
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  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + (0.222 − 0.974i)8-s + (−0.222 − 0.974i)10-s + (0.900 − 0.433i)11-s + (−0.900 + 0.433i)13-s + (−0.222 − 0.974i)16-s + (−0.623 − 0.781i)17-s + 19-s + (−0.623 − 0.781i)20-s + (0.623 − 0.781i)22-s + (−0.623 + 0.781i)23-s + (−0.900 − 0.433i)25-s + (−0.623 + 0.781i)26-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s + (0.222 − 0.974i)8-s + (−0.222 − 0.974i)10-s + (0.900 − 0.433i)11-s + (−0.900 + 0.433i)13-s + (−0.222 − 0.974i)16-s + (−0.623 − 0.781i)17-s + 19-s + (−0.623 − 0.781i)20-s + (0.623 − 0.781i)22-s + (−0.623 + 0.781i)23-s + (−0.900 − 0.433i)25-s + (−0.623 + 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(147\)    =    \(3 \cdot 7^{2}\)
Sign: $-0.462 - 0.886i$
Analytic conductor: \(15.7973\)
Root analytic conductor: \(15.7973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{147} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 147,\ (1:\ ),\ -0.462 - 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.599092409 - 2.637870187i\)
\(L(\frac12)\) \(\approx\) \(1.599092409 - 2.637870187i\)
\(L(1)\) \(\approx\) \(1.563559173 - 1.035782741i\)
\(L(1)\) \(\approx\) \(1.563559173 - 1.035782741i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (0.222 - 0.974i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (-0.623 - 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.623 + 0.781i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (-0.623 + 0.781i)T \)
59 \( 1 + (0.222 + 0.974i)T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 + T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 + T \)
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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

−28.41707450800003443583669212024, −26.86734051404304801258022167235, −26.2415082184702560289290699746, −25.097151402420817223797027009848, −24.4224949487360628217415174413, −23.157770350978658679923629756984, −22.21289167080335948669831258196, −21.903332538340154123107508824138, −20.40393495411929336547228283423, −19.51013160315279625155828438937, −17.96519656286569656748430547277, −17.19424808555266783696760075620, −15.91821241862510170499205915320, −14.74748022795431880178249241144, −14.35984834761135722090739922502, −13.04009046107538050876126820498, −11.97973219429765002752904966270, −10.92880233073259159422421575380, −9.64042186801856329432010624560, −7.93798419996321890755599435580, −6.89478807223548274418031475706, −6.03322855497322928585346983906, −4.60739527175773689483396819990, −3.35133030058109312833687895019, −2.13215132085697260312881748665, 0.88902845351441869281264737730, 2.28164873742631371571846383781, 3.89610059498076170343511855707, 4.92881443550561150559856270055, 5.9906478365915661610525353133, 7.34900543339697770699731678312, 9.09479218090493271257387196566, 9.92162366874562518293141493981, 11.63449423730782672630068761732, 12.03670939432252260117042999486, 13.46299205810516515137791369237, 14.00777734571314246394912655624, 15.391241675677682641475412494835, 16.37153733375941680059852993283, 17.410494952946015075093495696923, 18.996079395633952665121246704584, 19.95407151023763281640587528961, 20.64843034784239871200042912250, 21.81364505655487693049800095993, 22.4181561903771655826744743045, 23.81202722448504634526681231149, 24.50867783475692968306399099488, 25.14964454322502891947051110262, 26.82098930755348509028332906753, 27.87370044240249022677323495010

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