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Properties

Label 1-171-171.128-r0-0-0
Degree $1$
Conductor $171$
Sign $0.998 - 0.0540i$
Analytic cond. $0.794120$
Root an. cond. $0.794120$
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  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (0.939 + 0.342i)5-s + 7-s + (−0.5 − 0.866i)8-s + (−0.766 − 0.642i)10-s + (0.5 − 0.866i)11-s + (−0.173 + 0.984i)13-s + (−0.939 − 0.342i)14-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + (0.5 + 0.866i)20-s + (−0.766 + 0.642i)22-s + (−0.766 − 0.642i)23-s + (0.766 + 0.642i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (0.939 + 0.342i)5-s + 7-s + (−0.5 − 0.866i)8-s + (−0.766 − 0.642i)10-s + (0.5 − 0.866i)11-s + (−0.173 + 0.984i)13-s + (−0.939 − 0.342i)14-s + (0.173 + 0.984i)16-s + (−0.766 + 0.642i)17-s + (0.5 + 0.866i)20-s + (−0.766 + 0.642i)22-s + (−0.766 − 0.642i)23-s + (0.766 + 0.642i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.998 - 0.0540i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ 0.998 - 0.0540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9645633776 + 0.02611032014i\)
\(L(\frac12)\) \(\approx\) \(0.9645633776 + 0.02611032014i\)
\(L(1)\) \(\approx\) \(0.9017979350 - 0.04720519401i\)
\(L(1)\) \(\approx\) \(0.9017979350 - 0.04720519401i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.173 + 0.984i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (0.939 + 0.342i)T \)
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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

−27.79658730115915953281526587379, −26.57317517828528472569344844620, −25.55085213580398890372994676181, −24.79522362444173900091453699732, −24.19501348577161177057293974238, −22.80495174609184705403520580296, −21.52669594628277019743292732040, −20.43080174324087539890619809785, −19.95282171805460609248835910509, −18.3001159715422847235124318307, −17.68092442594150842979367571214, −17.13095468331928039872041988891, −15.78577709158802824365159591727, −14.79060410117791347280653200377, −13.846683515942523785007493904688, −12.36601437014854864395561947855, −11.166576106123841633917046127983, −10.08083862900809954325397793927, −9.22243340528333786710773017551, −8.14459283227473112591831741170, −7.054877369532321686391339134116, −5.7704097300355757699481508227, −4.76492468028576658864181771394, −2.40212875993002881703497692203, −1.34698723395357512679822850544, 1.4983454780685681266558948728, 2.46332485554940538922236859155, 4.155823877055796728438957008074, 5.95908169659581398564656471352, 6.93135877283636379993179665851, 8.363947022071212676875766018647, 9.118910263601267700290637407736, 10.36543042597911259240473346492, 11.15987362279123016182323974111, 12.15767741134650754289937905857, 13.68551254527397364359452434201, 14.53238005162911695235443048771, 15.94814234922821032396382723055, 17.14051218867630962447165243170, 17.640066917677546704316368303681, 18.69665215767792322129559304032, 19.53216850920036907477761649656, 20.88504637982777376695225801042, 21.42757811527983394406189260719, 22.29303997853143693390016011268, 24.25771581905216359644163419502, 24.57616050291130526777823626264, 25.93494477046579325616708951299, 26.564831736265773878031788901, 27.45990177017637137890509729995

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