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Properties

Label 1-177-177.92-r0-0-0
Degree $1$
Conductor $177$
Sign $0.998 + 0.0587i$
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  + (0.267 − 0.963i)2-s + (−0.856 − 0.515i)4-s + (−0.0541 + 0.998i)5-s + (−0.161 + 0.986i)7-s + (−0.725 + 0.687i)8-s + (0.947 + 0.319i)10-s + (0.468 − 0.883i)11-s + (0.370 + 0.928i)13-s + (0.907 + 0.419i)14-s + (0.468 + 0.883i)16-s + (0.161 + 0.986i)17-s + (0.647 + 0.762i)19-s + (0.561 − 0.827i)20-s + (−0.725 − 0.687i)22-s + (0.796 − 0.605i)23-s + ⋯
L(s)  = 1  + (0.267 − 0.963i)2-s + (−0.856 − 0.515i)4-s + (−0.0541 + 0.998i)5-s + (−0.161 + 0.986i)7-s + (−0.725 + 0.687i)8-s + (0.947 + 0.319i)10-s + (0.468 − 0.883i)11-s + (0.370 + 0.928i)13-s + (0.907 + 0.419i)14-s + (0.468 + 0.883i)16-s + (0.161 + 0.986i)17-s + (0.647 + 0.762i)19-s + (0.561 − 0.827i)20-s + (−0.725 − 0.687i)22-s + (0.796 − 0.605i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.998 + 0.0587i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ 0.998 + 0.0587i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.118932738 + 0.03289273292i\)
\(L(\frac12)\) \(\approx\) \(1.118932738 + 0.03289273292i\)
\(L(1)\) \(\approx\) \(1.064206456 - 0.1634791152i\)
\(L(1)\) \(\approx\) \(1.064206456 - 0.1634791152i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.267 - 0.963i)T \)
5 \( 1 + (-0.0541 + 0.998i)T \)
7 \( 1 + (-0.161 + 0.986i)T \)
11 \( 1 + (0.468 - 0.883i)T \)
13 \( 1 + (0.370 + 0.928i)T \)
17 \( 1 + (0.161 + 0.986i)T \)
19 \( 1 + (0.647 + 0.762i)T \)
23 \( 1 + (0.796 - 0.605i)T \)
29 \( 1 + (-0.267 - 0.963i)T \)
31 \( 1 + (-0.647 + 0.762i)T \)
37 \( 1 + (0.725 + 0.687i)T \)
41 \( 1 + (-0.796 - 0.605i)T \)
43 \( 1 + (-0.468 - 0.883i)T \)
47 \( 1 + (0.0541 + 0.998i)T \)
53 \( 1 + (0.947 - 0.319i)T \)
61 \( 1 + (-0.267 + 0.963i)T \)
67 \( 1 + (0.725 - 0.687i)T \)
71 \( 1 + (-0.0541 - 0.998i)T \)
73 \( 1 + (-0.907 - 0.419i)T \)
79 \( 1 + (-0.561 + 0.827i)T \)
83 \( 1 + (0.976 + 0.214i)T \)
89 \( 1 + (0.267 + 0.963i)T \)
97 \( 1 + (-0.907 + 0.419i)T \)
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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

−27.381883623429501094528240953905, −26.2717931122923648490624807496, −25.241365408829594270029435134735, −24.66130744034915103474504018176, −23.44678943287951670294663801165, −23.00819288667884185430928328073, −21.82384732693556080889562754893, −20.45821774128543502747533728175, −19.9587758264429470250247271469, −18.22213017061746001223135993512, −17.3534112181832513004742458495, −16.57031346480638257575381013700, −15.71097011145729958512361287407, −14.63195238220801492499397594350, −13.37561568629590768828417582771, −12.919188990941995711098868231398, −11.57525027695043041401691617239, −9.83960097400664410999094411246, −9.02625817294931304346258885304, −7.70309826704900319783649574191, −6.9907791549064708509563258333, −5.44329967203908714632811842079, −4.60381538560888741838362381096, −3.43284347004782550611239497598, −0.92571591910869464871397147897, 1.742050506828204892364836750212, 3.004656966708991168456288257494, 3.913850904782040254957917955323, 5.61513049721449259370886726287, 6.48506733858171646032073480806, 8.35180726643666252894220193198, 9.325831655541140515725530487121, 10.49922350249039218936819671662, 11.438745597911610203740174311940, 12.17627092836825115778689334523, 13.51430746015817551380161874062, 14.42466506777671390894157550539, 15.235240467572988525635032472033, 16.70990256243594988136154602889, 18.160982772112172898245761611393, 18.88987535924019155083598492333, 19.341770503196154019596834108140, 20.86652550895560412341384283250, 21.73741584326742391853033989388, 22.27745721655024870989548000411, 23.30103464126904172911351502637, 24.3543486909590826479283754563, 25.635140579979593028684936676916, 26.72454294951014487491221526103, 27.40682687569300703950414690616

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