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Properties

Label 1-177-177.32-r0-0-0
Degree $1$
Conductor $177$
Sign $0.449 - 0.893i$
Analytic cond. $0.821984$
Root an. cond. $0.821984$
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  + (−0.856 − 0.515i)2-s + (0.468 + 0.883i)4-s + (0.994 + 0.108i)5-s + (−0.947 − 0.319i)7-s + (0.0541 − 0.998i)8-s + (−0.796 − 0.605i)10-s + (−0.561 − 0.827i)11-s + (0.725 − 0.687i)13-s + (0.647 + 0.762i)14-s + (−0.561 + 0.827i)16-s + (0.947 − 0.319i)17-s + (−0.161 + 0.986i)19-s + (0.370 + 0.928i)20-s + (0.0541 + 0.998i)22-s + (0.267 − 0.963i)23-s + ⋯
L(s)  = 1  + (−0.856 − 0.515i)2-s + (0.468 + 0.883i)4-s + (0.994 + 0.108i)5-s + (−0.947 − 0.319i)7-s + (0.0541 − 0.998i)8-s + (−0.796 − 0.605i)10-s + (−0.561 − 0.827i)11-s + (0.725 − 0.687i)13-s + (0.647 + 0.762i)14-s + (−0.561 + 0.827i)16-s + (0.947 − 0.319i)17-s + (−0.161 + 0.986i)19-s + (0.370 + 0.928i)20-s + (0.0541 + 0.998i)22-s + (0.267 − 0.963i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.449 - 0.893i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ 0.449 - 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6976402696 - 0.4298752939i\)
\(L(\frac12)\) \(\approx\) \(0.6976402696 - 0.4298752939i\)
\(L(1)\) \(\approx\) \(0.7567497619 - 0.2497508000i\)
\(L(1)\) \(\approx\) \(0.7567497619 - 0.2497508000i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.856 - 0.515i)T \)
5 \( 1 + (0.994 + 0.108i)T \)
7 \( 1 + (-0.947 - 0.319i)T \)
11 \( 1 + (-0.561 - 0.827i)T \)
13 \( 1 + (0.725 - 0.687i)T \)
17 \( 1 + (0.947 - 0.319i)T \)
19 \( 1 + (-0.161 + 0.986i)T \)
23 \( 1 + (0.267 - 0.963i)T \)
29 \( 1 + (0.856 - 0.515i)T \)
31 \( 1 + (0.161 + 0.986i)T \)
37 \( 1 + (-0.0541 - 0.998i)T \)
41 \( 1 + (-0.267 - 0.963i)T \)
43 \( 1 + (0.561 - 0.827i)T \)
47 \( 1 + (-0.994 + 0.108i)T \)
53 \( 1 + (-0.796 + 0.605i)T \)
61 \( 1 + (0.856 + 0.515i)T \)
67 \( 1 + (-0.0541 + 0.998i)T \)
71 \( 1 + (0.994 - 0.108i)T \)
73 \( 1 + (-0.647 - 0.762i)T \)
79 \( 1 + (-0.370 - 0.928i)T \)
83 \( 1 + (0.907 + 0.419i)T \)
89 \( 1 + (-0.856 + 0.515i)T \)
97 \( 1 + (-0.647 + 0.762i)T \)
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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

−27.71907177732839117188606727521, −26.0435064665964312941481728855, −25.89715960768855190752909466492, −25.09581501062442945429527871632, −23.85100924397734528066171217410, −23.01174736698615214502475657201, −21.656710892337481212617606294647, −20.72880152520261613463207843202, −19.569929065964107179528408903916, −18.6374118554568909769578694536, −17.80098716246664925896201765125, −16.85991178811625144800327722726, −15.95206031410750521094534297406, −14.99840874418068640700324214932, −13.75422472222313385592430829534, −12.77581619262952447771234627019, −11.26363432345121663277607064207, −9.9157777593965435949073409382, −9.53773940949575106302658125132, −8.33037170157809802003756047703, −6.89917764010986761436517692333, −6.12172147886670692554965116346, −4.99737557422244241305124838186, −2.81884339864472748969895173924, −1.49377469973336780611593792175, 0.97504749362345392336404027258, 2.63689086061237361746091679394, 3.54476575882941235599845259878, 5.67798951171091037823256213539, 6.66484261183995864629095032195, 8.05240360367766152298847121195, 9.09178763095131227296606837380, 10.293710624948400205559789627493, 10.58869548515133381502079059852, 12.28926478718212660243454284812, 13.12691506299257365335461312712, 14.117718786621480595116337716674, 15.91820510420598908758824634969, 16.53395779676560401247184678682, 17.598129690326887079406719550404, 18.57134722824678421409886966204, 19.22274992603710049058422699736, 20.61594073567577461649643431774, 21.10142326156426005833315290574, 22.23735170582603519751966677216, 23.197790659954013831949089454671, 24.89077639322432978742648725876, 25.445005591452497160466066791542, 26.324968864078165150201745923729, 27.14086426669980511965003712034

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