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Properties

Label 1-171-171.92-r1-0-0
Degree $1$
Conductor $171$
Sign $0.982 + 0.188i$
Analytic cond. $18.3765$
Root an. cond. $18.3765$
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  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.766 + 0.642i)5-s + 7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)10-s + (0.5 − 0.866i)11-s + (−0.939 + 0.342i)13-s + (−0.766 + 0.642i)14-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + (0.5 + 0.866i)20-s + (0.173 + 0.984i)22-s + (−0.173 + 0.984i)23-s + (0.173 − 0.984i)25-s + (0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (−0.766 + 0.642i)5-s + 7-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)10-s + (0.5 − 0.866i)11-s + (−0.939 + 0.342i)13-s + (−0.766 + 0.642i)14-s + (−0.939 − 0.342i)16-s + (−0.173 − 0.984i)17-s + (0.5 + 0.866i)20-s + (0.173 + 0.984i)22-s + (−0.173 + 0.984i)23-s + (0.173 − 0.984i)25-s + (0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.982 + 0.188i$
Analytic conductor: \(18.3765\)
Root analytic conductor: \(18.3765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (1:\ ),\ 0.982 + 0.188i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.078818120 + 0.1026261622i\)
\(L(\frac12)\) \(\approx\) \(1.078818120 + 0.1026261622i\)
\(L(1)\) \(\approx\) \(0.7424063332 + 0.1627210257i\)
\(L(1)\) \(\approx\) \(0.7424063332 + 0.1627210257i\)

Euler product

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

−27.29337681521551845647651365834, −26.79382683169409259169967215882, −25.35777947056498656434501611950, −24.544938872176406709109556202170, −23.49195578049699453950455314758, −22.18769886112138880271156439752, −21.230166068324329152577477702440, −20.07554851105814800913323976347, −19.83028151875381244622551914857, −18.47486904390164354814721587688, −17.4390111953109383977052303997, −16.81170038496403363657581961768, −15.50150885566971394838051242484, −14.47597396034829671936706417384, −12.68132289720270529291010075308, −12.18421279229545311936308845066, −11.122674326859704870169822043943, −10.05373932813490693481260937299, −8.76632228058743717581007773138, −8.02605843933247824776468021738, −7.00714292770080692385102621091, −4.86166792952123947402953568459, −3.97993135293703866614682559750, −2.247140572262257822986681575815, −0.97202676895212376304702834247, 0.68238992017842971100302313097, 2.41985777104279279757156882111, 4.26218550655623604218131063359, 5.567853723054761817217551644902, 6.9654355417679154847036143318, 7.716557073611085049820327097544, 8.735128284261225285598117999930, 9.95153585561971846313156658693, 11.279556546897200518272965341889, 11.677512048996446588198533830200, 13.89051520732060676282165328987, 14.57798876670626465750208983810, 15.49954082789621053324395664905, 16.51991408546920238307709791521, 17.5484008513447650567048567410, 18.46638437557472747205726666351, 19.33102864267257390465516113216, 20.14884366946453288643344411602, 21.578880765445150677623960759137, 22.70817069169390300692266789964, 23.834330398020864007858736666487, 24.379172347795058136496728063719, 25.432785476508353955969644632020, 26.75717607886900648326644435777, 27.07593605457939537948287836835

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