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Properties

Label 1-177-177.167-r1-0-0
Degree $1$
Conductor $177$
Sign $-0.893 + 0.449i$
Analytic cond. $19.0212$
Root an. cond. $19.0212$
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  + (0.725 + 0.687i)2-s + (0.0541 + 0.998i)4-s + (0.161 + 0.986i)5-s + (0.468 + 0.883i)7-s + (−0.647 + 0.762i)8-s + (−0.561 + 0.827i)10-s + (0.994 + 0.108i)11-s + (0.907 − 0.419i)13-s + (−0.267 + 0.963i)14-s + (−0.994 + 0.108i)16-s + (−0.468 + 0.883i)17-s + (−0.856 − 0.515i)19-s + (−0.976 + 0.214i)20-s + (0.647 + 0.762i)22-s + (0.370 − 0.928i)23-s + ⋯
L(s)  = 1  + (0.725 + 0.687i)2-s + (0.0541 + 0.998i)4-s + (0.161 + 0.986i)5-s + (0.468 + 0.883i)7-s + (−0.647 + 0.762i)8-s + (−0.561 + 0.827i)10-s + (0.994 + 0.108i)11-s + (0.907 − 0.419i)13-s + (−0.267 + 0.963i)14-s + (−0.994 + 0.108i)16-s + (−0.468 + 0.883i)17-s + (−0.856 − 0.515i)19-s + (−0.976 + 0.214i)20-s + (0.647 + 0.762i)22-s + (0.370 − 0.928i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.893 + 0.449i$
Analytic conductor: \(19.0212\)
Root analytic conductor: \(19.0212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (1:\ ),\ -0.893 + 0.449i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6682606811 + 2.817652610i\)
\(L(\frac12)\) \(\approx\) \(0.6682606811 + 2.817652610i\)
\(L(1)\) \(\approx\) \(1.218647573 + 1.226616096i\)
\(L(1)\) \(\approx\) \(1.218647573 + 1.226616096i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.725 + 0.687i)T \)
5 \( 1 + (0.161 + 0.986i)T \)
7 \( 1 + (0.468 + 0.883i)T \)
11 \( 1 + (0.994 + 0.108i)T \)
13 \( 1 + (0.907 - 0.419i)T \)
17 \( 1 + (-0.468 + 0.883i)T \)
19 \( 1 + (-0.856 - 0.515i)T \)
23 \( 1 + (0.370 - 0.928i)T \)
29 \( 1 + (0.725 - 0.687i)T \)
31 \( 1 + (-0.856 + 0.515i)T \)
37 \( 1 + (0.647 + 0.762i)T \)
41 \( 1 + (0.370 + 0.928i)T \)
43 \( 1 + (-0.994 + 0.108i)T \)
47 \( 1 + (0.161 - 0.986i)T \)
53 \( 1 + (0.561 + 0.827i)T \)
61 \( 1 + (-0.725 - 0.687i)T \)
67 \( 1 + (0.647 - 0.762i)T \)
71 \( 1 + (0.161 - 0.986i)T \)
73 \( 1 + (0.267 - 0.963i)T \)
79 \( 1 + (0.976 - 0.214i)T \)
83 \( 1 + (-0.796 + 0.605i)T \)
89 \( 1 + (0.725 - 0.687i)T \)
97 \( 1 + (0.267 + 0.963i)T \)
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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

−27.3027535758546321742855960444, −25.52251986763177787011038982014, −24.57794793483892096389589563594, −23.70008159412101656191435572513, −23.02900397759824535362105492841, −21.723190093119111723607116582380, −20.909992007221209758932304289, −20.17978280813695316775418552809, −19.38666649511184888191750202617, −18.01965137351326812931507304390, −16.87121722405020811468408561828, −15.91752891624014705403737741659, −14.49155925214500949006662643874, −13.71645140191517668389456785026, −12.879226633852428231023814540589, −11.69733074882816825142996617264, −10.93316837798645653181335441370, −9.58652494685722354130220335404, −8.66556829031780252821832827661, −6.95414694916436185687883674695, −5.69355381297220481475872119753, −4.470998262888396745841068560823, −3.75600826111107670966550389805, −1.800644526851275111827847756461, −0.8517794659727025474327035634, 2.134409079474608167630510900822, 3.383766619929913266770174931069, 4.62255905177130154998379035396, 6.13262932690760846985538809033, 6.56567117514535014410645459084, 8.097449158809592081702730600238, 8.97275573084900077282301243516, 10.75105186044010907344556188510, 11.65368820839981042764701244951, 12.79351274076288689939343799263, 13.897816680919808093414256155843, 14.99855917129329310721585140277, 15.22993723519544223212783766868, 16.74325215045364730141947429556, 17.755125102962905819419554850848, 18.527420462027632776275419917462, 19.89289132052195096059903413877, 21.352931743214321094671253617214, 21.84094251052423856838645850474, 22.7735484585837029589864264552, 23.63954805740851843978068439805, 24.86716729539605125168870044253, 25.399941645071851058199771441266, 26.37704385597209477078240768441, 27.35681183926842370682022485433

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