Properties

Label 1-171-171.155-r0-0-0
Degree $1$
Conductor $171$
Sign $0.342 - 0.939i$
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  + (0.766 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(171\)    =    \(3^{2} \cdot 19\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(0.794120\)
Root analytic conductor: \(0.794120\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{171} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 171,\ (0:\ ),\ 0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.357056461 - 0.9495612988i\)
\(L(\frac12)\) \(\approx\) \(1.357056461 - 0.9495612988i\)
\(L(1)\) \(\approx\) \(1.376153636 - 0.6056063303i\)
\(L(1)\) \(\approx\) \(1.376153636 - 0.6056063303i\)

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

−27.744562129989912801841639707269, −26.62055418968528065574629185593, −25.651409699493303622031553221178, −24.492829906632052715571512038519, −23.98582279700736885439175972399, −23.091890277535588187436555636848, −22.1318406601743002348357369372, −20.736674666190195347144358845159, −20.51155776521335586233215785561, −18.92304362781611501142697014029, −17.52257924243628417841041065439, −16.84804427939838776652638376213, −15.641772647480226556899841338614, −14.95995767722747536067169578267, −13.92085950374331046670488977020, −12.70689116041798660841774321383, −11.91375617307475979615870218152, −10.93577595279507941991924012392, −8.899131730134924746462857850238, −8.14293723711984640236397941602, −7.09681509382089177592727682604, −5.75157009337314293918813978726, −4.463300254330169203957107602525, −3.91040522717022155789647937787, −1.85906026772573875324039020867, 1.30881767276854106418901644674, 2.99238122829372759168572597001, 3.91396157017113521919476457783, 5.17404744276052881399558565612, 6.4169660164466431228479516398, 7.735059370933505164305914127837, 9.05790811121924400730446086326, 10.686631289857016977133026137281, 11.28911045902658175450289273802, 12.01393998936403409811558206667, 13.55822445284451127451309947125, 14.258988089463422493022020896668, 15.26261279503384039826997460658, 16.13606269145358934517871516776, 17.9083172864801379026474205879, 18.72319234209220820663547028834, 19.69491673839285423054572568140, 20.6368240689858886900022130222, 21.56695360128226636210930106700, 22.51795569249310757869648257788, 23.38463708078890726988409647815, 24.16286649309545394696567916619, 25.15432632220347230468204981582, 26.73380183535226067041384584377, 27.49181604576149403172855587167

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