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Properties

Label 1-547-547.525-r0-0-0
Degree $1$
Conductor $547$
Sign $0.931 - 0.364i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
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  + (−0.684 − 0.729i)2-s + (−0.222 + 0.974i)3-s + (−0.0632 + 0.997i)4-s + (0.166 − 0.986i)5-s + (0.863 − 0.504i)6-s + (0.785 − 0.618i)7-s + (0.770 − 0.636i)8-s + (−0.900 − 0.433i)9-s + (−0.832 + 0.553i)10-s + (0.799 + 0.600i)11-s + (−0.958 − 0.283i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (0.924 + 0.381i)15-s + (−0.991 − 0.126i)16-s + (0.658 + 0.752i)17-s + ⋯
L(s)  = 1  + (−0.684 − 0.729i)2-s + (−0.222 + 0.974i)3-s + (−0.0632 + 0.997i)4-s + (0.166 − 0.986i)5-s + (0.863 − 0.504i)6-s + (0.785 − 0.618i)7-s + (0.770 − 0.636i)8-s + (−0.900 − 0.433i)9-s + (−0.832 + 0.553i)10-s + (0.799 + 0.600i)11-s + (−0.958 − 0.283i)12-s + (0.365 + 0.930i)13-s + (−0.988 − 0.149i)14-s + (0.924 + 0.381i)15-s + (−0.991 − 0.126i)16-s + (0.658 + 0.752i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.931 - 0.364i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (525, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.931 - 0.364i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.035795779 - 0.1954937747i\)
\(L(\frac12)\) \(\approx\) \(1.035795779 - 0.1954937747i\)
\(L(1)\) \(\approx\) \(0.8459729022 - 0.1233410207i\)
\(L(1)\) \(\approx\) \(0.8459729022 - 0.1233410207i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.684 - 0.729i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.166 - 0.986i)T \)
7 \( 1 + (0.785 - 0.618i)T \)
11 \( 1 + (0.799 + 0.600i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (0.658 + 0.752i)T \)
19 \( 1 + (0.838 - 0.544i)T \)
23 \( 1 + (-0.806 - 0.591i)T \)
29 \( 1 + (-0.596 + 0.802i)T \)
31 \( 1 + (0.813 - 0.582i)T \)
37 \( 1 + (0.469 - 0.882i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.614 + 0.788i)T \)
47 \( 1 + (-0.845 - 0.534i)T \)
53 \( 1 + (-0.131 - 0.991i)T \)
59 \( 1 + (0.948 - 0.316i)T \)
61 \( 1 + (0.932 + 0.359i)T \)
67 \( 1 + (0.233 + 0.972i)T \)
71 \( 1 + (-0.244 + 0.969i)T \)
73 \( 1 + (0.924 - 0.381i)T \)
79 \( 1 + (0.990 + 0.137i)T \)
83 \( 1 + (0.692 - 0.721i)T \)
89 \( 1 + (-0.154 + 0.987i)T \)
97 \( 1 + (0.0287 - 0.999i)T \)
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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

−23.63182854017465029144847378279, −22.687898646728979052033476203316, −22.20525718745475537519227988652, −20.7423107606834967355613715413, −19.70536299155129984953766391650, −18.79905638131022386012334690424, −18.37165175788815816617539476957, −17.7150988497265657916043363788, −16.98301346704344200625944958158, −15.82160663081641158675129559943, −14.92578629454813532843414183906, −14.054535790767894143343239251681, −13.66930588565631413421966032486, −11.91909417782185287004565499554, −11.45646514729339426847620916482, −10.42887438152241747538211270188, −9.39030848993538240144185785892, −8.14689182550470186903055990608, −7.77256987523192724570898314545, −6.69376617943460927660294584992, −5.88511764208526307583594672560, −5.29598300414526321416984493547, −3.28975436723121780709976404024, −2.05013684107999650333025649501, −1.0282477688862445972698215041, 1.00175022815397013624430929537, 1.96164422540196550212944176979, 3.69611744013934382516094473833, 4.27562949331778767904050285972, 5.11736068507616982899336094723, 6.60963837354564234848509934831, 7.97984193635685681516440460800, 8.70471515467887310284042397030, 9.60767569137711683703547116798, 10.12224039658795191678130677406, 11.412954135814445844410885168674, 11.698397798401889276369665477970, 12.8379112311662568720703763763, 13.95107324809218021259665224552, 14.826502893972438142155347631776, 16.34330437556411828946589415307, 16.515519626531741967789160964095, 17.445822118299609195729936295041, 18.01503922003012951777103038975, 19.48059501514651796020620574695, 20.194977483102085723165146887604, 20.7163495345970335084603060904, 21.42981450168896691380188253079, 22.12123379836568970929913517319, 23.21936268821986549688916799831

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