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Properties

Label 1-177-177.8-r0-0-0
Degree $1$
Conductor $177$
Sign $0.213 - 0.977i$
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  + (−0.947 − 0.319i)2-s + (0.796 + 0.605i)4-s + (0.370 − 0.928i)5-s + (0.907 − 0.419i)7-s + (−0.561 − 0.827i)8-s + (−0.647 + 0.762i)10-s + (0.267 − 0.963i)11-s + (−0.468 + 0.883i)13-s + (−0.994 + 0.108i)14-s + (0.267 + 0.963i)16-s + (−0.907 − 0.419i)17-s + (0.976 − 0.214i)19-s + (0.856 − 0.515i)20-s + (−0.561 + 0.827i)22-s + (−0.161 + 0.986i)23-s + ⋯
L(s)  = 1  + (−0.947 − 0.319i)2-s + (0.796 + 0.605i)4-s + (0.370 − 0.928i)5-s + (0.907 − 0.419i)7-s + (−0.561 − 0.827i)8-s + (−0.647 + 0.762i)10-s + (0.267 − 0.963i)11-s + (−0.468 + 0.883i)13-s + (−0.994 + 0.108i)14-s + (0.267 + 0.963i)16-s + (−0.907 − 0.419i)17-s + (0.976 − 0.214i)19-s + (0.856 − 0.515i)20-s + (−0.561 + 0.827i)22-s + (−0.161 + 0.986i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.213 - 0.977i$
Analytic conductor: \(0.821984\)
Root analytic conductor: \(0.821984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (0:\ ),\ 0.213 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6554159081 - 0.5278719910i\)
\(L(\frac12)\) \(\approx\) \(0.6554159081 - 0.5278719910i\)
\(L(1)\) \(\approx\) \(0.7514646019 - 0.3028643275i\)
\(L(1)\) \(\approx\) \(0.7514646019 - 0.3028643275i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 \)
good2 \( 1 + (-0.947 - 0.319i)T \)
5 \( 1 + (0.370 - 0.928i)T \)
7 \( 1 + (0.907 - 0.419i)T \)
11 \( 1 + (0.267 - 0.963i)T \)
13 \( 1 + (-0.468 + 0.883i)T \)
17 \( 1 + (-0.907 - 0.419i)T \)
19 \( 1 + (0.976 - 0.214i)T \)
23 \( 1 + (-0.161 + 0.986i)T \)
29 \( 1 + (0.947 - 0.319i)T \)
31 \( 1 + (-0.976 - 0.214i)T \)
37 \( 1 + (0.561 - 0.827i)T \)
41 \( 1 + (0.161 + 0.986i)T \)
43 \( 1 + (-0.267 - 0.963i)T \)
47 \( 1 + (-0.370 - 0.928i)T \)
53 \( 1 + (-0.647 - 0.762i)T \)
61 \( 1 + (0.947 + 0.319i)T \)
67 \( 1 + (0.561 + 0.827i)T \)
71 \( 1 + (0.370 + 0.928i)T \)
73 \( 1 + (0.994 - 0.108i)T \)
79 \( 1 + (-0.856 + 0.515i)T \)
83 \( 1 + (0.0541 + 0.998i)T \)
89 \( 1 + (-0.947 + 0.319i)T \)
97 \( 1 + (0.994 + 0.108i)T \)
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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

−27.3755064500520472777354178080, −26.773177175558078253138951572930, −25.686399620867217639921601838032, −24.96097378942391334532960304435, −24.11227012800135072896414184135, −22.78414046709317046031388146114, −21.85790563468618384468154038741, −20.56177247173144143258905959973, −19.79902957386024282995652535958, −18.466939898759483791260885279676, −17.8939169789008183885188870437, −17.25684745167841757199866129024, −15.71651679515791419847685392553, −14.88085426900385471110909603560, −14.24465382644568532774769549668, −12.41968234654144778277647172486, −11.23861827081367030792091562725, −10.39068609581130346515070879539, −9.43086642217252890443123709537, −8.1578828171594556813595021387, −7.22700046149291316366638219080, −6.16722792444117676005138748335, −4.90999444144271863733284409632, −2.768501765905252919058892668856, −1.71440237489888913958873729228, 0.99902186517014432157402310062, 2.17283195302696197219324506507, 3.95220153679926823179592152277, 5.32651476958191018589874601823, 6.8593919126496455779525802960, 8.03736070307352506685531012754, 8.959680816457405906240887365404, 9.78446532865973986862980486715, 11.26011205152184558179160894501, 11.756682940302871521046598297479, 13.22115331761006843695129793960, 14.191989877882553381308424459875, 15.82301544176845501191573973183, 16.62788415906081352329207934997, 17.45849225417787933875301424649, 18.25933364467298790391414214649, 19.61000102467638362642223773445, 20.22208736725637656587080638758, 21.30572808809347796811759393333, 21.838185159452000814165405757211, 23.840945186101171526450463745386, 24.42656361894890971634098213951, 25.23399741246523476361644114416, 26.656202630573142291801873712414, 27.05024268140482153710709911771

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