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Properties

Label 1-177-177.173-r0-0-0
Degree $1$
Conductor $177$
Sign $0.932 - 0.360i$
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  + (0.976 + 0.214i)2-s + (0.907 + 0.419i)4-s + (−0.267 − 0.963i)5-s + (−0.725 − 0.687i)7-s + (0.796 + 0.605i)8-s + (−0.0541 − 0.998i)10-s + (0.647 − 0.762i)11-s + (0.947 + 0.319i)13-s + (−0.561 − 0.827i)14-s + (0.647 + 0.762i)16-s + (0.725 − 0.687i)17-s + (−0.370 + 0.928i)19-s + (0.161 − 0.986i)20-s + (0.796 − 0.605i)22-s + (−0.994 − 0.108i)23-s + ⋯
L(s)  = 1  + (0.976 + 0.214i)2-s + (0.907 + 0.419i)4-s + (−0.267 − 0.963i)5-s + (−0.725 − 0.687i)7-s + (0.796 + 0.605i)8-s + (−0.0541 − 0.998i)10-s + (0.647 − 0.762i)11-s + (0.947 + 0.319i)13-s + (−0.561 − 0.827i)14-s + (0.647 + 0.762i)16-s + (0.725 − 0.687i)17-s + (−0.370 + 0.928i)19-s + (0.161 − 0.986i)20-s + (0.796 − 0.605i)22-s + (−0.994 − 0.108i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.932 - 0.360i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ 0.932 - 0.360i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.895804623 - 0.3534441485i\)
\(L(\frac12)\) \(\approx\) \(1.895804623 - 0.3534441485i\)
\(L(1)\) \(\approx\) \(1.710256940 - 0.1336240794i\)
\(L(1)\) \(\approx\) \(1.710256940 - 0.1336240794i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (0.976 + 0.214i)T \)
5 \( 1 + (-0.267 - 0.963i)T \)
7 \( 1 + (-0.725 - 0.687i)T \)
11 \( 1 + (0.647 - 0.762i)T \)
13 \( 1 + (0.947 + 0.319i)T \)
17 \( 1 + (0.725 - 0.687i)T \)
19 \( 1 + (-0.370 + 0.928i)T \)
23 \( 1 + (-0.994 - 0.108i)T \)
29 \( 1 + (-0.976 + 0.214i)T \)
31 \( 1 + (0.370 + 0.928i)T \)
37 \( 1 + (-0.796 + 0.605i)T \)
41 \( 1 + (0.994 - 0.108i)T \)
43 \( 1 + (-0.647 - 0.762i)T \)
47 \( 1 + (0.267 - 0.963i)T \)
53 \( 1 + (-0.0541 + 0.998i)T \)
61 \( 1 + (-0.976 - 0.214i)T \)
67 \( 1 + (-0.796 - 0.605i)T \)
71 \( 1 + (-0.267 + 0.963i)T \)
73 \( 1 + (0.561 + 0.827i)T \)
79 \( 1 + (-0.161 + 0.986i)T \)
83 \( 1 + (0.468 - 0.883i)T \)
89 \( 1 + (0.976 - 0.214i)T \)
97 \( 1 + (0.561 - 0.827i)T \)
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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

−27.928922978493455510389959153209, −25.92713791041244561917719655790, −25.7367340579035553133858864096, −24.443932475509323197542535761186, −23.2923024002777595824728185377, −22.58962894669902885556136577171, −21.96157708931938506187258221980, −20.884037669966975088391881853854, −19.647488653480762764640304178998, −19.040171851690450506295201100976, −17.832830493477251826824263696960, −16.282980256150620043923940543182, −15.30542210829560390415509419303, −14.74713406468898853155620779621, −13.52550068352223162053775241328, −12.51115307333100088399490757412, −11.591265049319247288454800695269, −10.58001165966593936675627932148, −9.49291844358802149594308790750, −7.72416437699036917863387987435, −6.49372193956190446180190479036, −5.859777077584770097979425950467, −4.13076599420227832392946580395, −3.22484081427297556786145304788, −2.01393463825720129374376341634, 1.39240277422317644102374603840, 3.451798971290670970842571999753, 4.09003580293990495210809946237, 5.52661044723151725068918386518, 6.48087830322511578861539050044, 7.76033361955726471134782629285, 8.89413850424251589072597868560, 10.3867697081321235427889885422, 11.67638395135833698099308218519, 12.46911435393750324747664657919, 13.56651910570089295418521420846, 14.1570313137848434638355404623, 15.72452923082943768406211525950, 16.43713137037435315537941461165, 16.9589179842627621710483863057, 18.83199493021412117343651415828, 19.93567890901191324445411076335, 20.63861404946581762106840083507, 21.564549395429752429881663245911, 22.78516875461120433006566638075, 23.4243781227577643085279327948, 24.33033619545015068911758296363, 25.2108410710549150465971269130, 26.120393696807444092175649440695, 27.3535548121676753585595276845

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