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Properties

Label 1-177-177.35-r1-0-0
Degree $1$
Conductor $177$
Sign $-0.341 - 0.940i$
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  + (0.856 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.341 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.341 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.341 - 0.940i$
Analytic conductor: \(19.0212\)
Root analytic conductor: \(19.0212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 177,\ (1:\ ),\ -0.341 - 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.810380342 - 2.582856447i\)
\(L(\frac12)\) \(\approx\) \(1.810380342 - 2.582856447i\)
\(L(1)\) \(\approx\) \(1.611915537 - 0.9551453976i\)
\(L(1)\) \(\approx\) \(1.611915537 - 0.9551453976i\)

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

−27.235377996721428496674979275303, −26.00410368555577463590981524866, −25.494494911003233955716574591577, −24.71782419986013618443886269242, −23.46054080727776147787014579316, −22.64553660744667944532780124210, −21.87668240278670488326102210871, −20.95144762899619601201410770558, −19.94127170833662593136384170052, −18.63890357978643915710360655527, −17.12596283753026916155656832006, −16.85792103082717681310731686975, −15.53604834815721823835517099499, −14.3792289493831323668630022829, −13.7626241282839471585267595450, −12.64471633318693774514721492177, −11.85756163148823109149437419715, −10.11829988007924338992526893600, −9.38104297644902707406199831526, −7.6250898129194379835442297132, −6.64186945601922573689648175509, −5.80099006507237354328723801540, −4.49780005138393537900584739208, −3.2330343921949459785183449027, −1.89071106299417345395026344479, 0.84393933657699187301811399123, 2.4676073185931866814152120068, 3.36548763534915352660403008353, 5.00759680840319883026056356916, 5.958384661028306116834470310308, 6.801892817651423576083516145490, 8.84861135432393440968999506306, 9.90364901000497810088570796959, 10.68943028919937679032795571396, 12.20322643999692870531028092769, 12.84085927227119703354448510796, 13.8808487063626909026968412654, 14.68600649344463939219947727104, 15.95700146040618749449506228436, 16.92577541677628224716867347409, 18.29405241951220207245966140912, 19.36854980618032674571251041755, 20.1017508943088273529980201640, 21.45390632630518239571003470445, 21.89451845822678262642753240200, 22.73897201336416366059391739202, 23.94516557431138053686718853777, 24.949346380640121674424119954362, 25.51846600730417058778959724327, 26.936441775362920519322168067223

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