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Properties

Label 1-547-547.86-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.902 + 0.430i$
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  + (−0.109 + 0.994i)2-s + (−0.222 − 0.974i)3-s + (−0.976 − 0.216i)4-s + (0.658 − 0.752i)5-s + (0.993 − 0.114i)6-s + (−0.965 − 0.261i)7-s + (0.322 − 0.946i)8-s + (−0.900 + 0.433i)9-s + (0.675 + 0.736i)10-s + (−0.200 − 0.979i)11-s + (0.00575 + 0.999i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (−0.880 − 0.474i)15-s + (0.905 + 0.423i)16-s + (−0.717 + 0.696i)17-s + ⋯
L(s)  = 1  + (−0.109 + 0.994i)2-s + (−0.222 − 0.974i)3-s + (−0.976 − 0.216i)4-s + (0.658 − 0.752i)5-s + (0.993 − 0.114i)6-s + (−0.965 − 0.261i)7-s + (0.322 − 0.946i)8-s + (−0.900 + 0.433i)9-s + (0.675 + 0.736i)10-s + (−0.200 − 0.979i)11-s + (0.00575 + 0.999i)12-s + (−0.988 + 0.149i)13-s + (0.365 − 0.930i)14-s + (−0.880 − 0.474i)15-s + (0.905 + 0.423i)16-s + (−0.717 + 0.696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.902 + 0.430i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.902 + 0.430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009123538629 + 0.04027715761i\)
\(L(\frac12)\) \(\approx\) \(0.009123538629 + 0.04027715761i\)
\(L(1)\) \(\approx\) \(0.6016548162 + 0.006230007404i\)
\(L(1)\) \(\approx\) \(0.6016548162 + 0.006230007404i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.109 + 0.994i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (0.658 - 0.752i)T \)
7 \( 1 + (-0.965 - 0.261i)T \)
11 \( 1 + (-0.200 - 0.979i)T \)
13 \( 1 + (-0.988 + 0.149i)T \)
17 \( 1 + (-0.717 + 0.696i)T \)
19 \( 1 + (-0.999 - 0.0115i)T \)
23 \( 1 + (-0.577 + 0.816i)T \)
29 \( 1 + (0.978 + 0.205i)T \)
31 \( 1 + (0.990 + 0.137i)T \)
37 \( 1 + (0.166 + 0.986i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.143 - 0.989i)T \)
47 \( 1 + (0.692 + 0.721i)T \)
53 \( 1 + (0.0287 + 0.999i)T \)
59 \( 1 + (-0.632 - 0.774i)T \)
61 \( 1 + (0.529 - 0.848i)T \)
67 \( 1 + (0.863 - 0.504i)T \)
71 \( 1 + (-0.958 + 0.283i)T \)
73 \( 1 + (-0.880 + 0.474i)T \)
79 \( 1 + (-0.928 + 0.370i)T \)
83 \( 1 + (0.799 + 0.600i)T \)
89 \( 1 + (-0.999 + 0.0345i)T \)
97 \( 1 + (-0.890 + 0.454i)T \)
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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

−22.68095534185468899776884644147, −22.11062637036692039557285934892, −21.46757930963713407431560856059, −20.59804533847964775955599454215, −19.77244138816442818531264030076, −19.00524846040840929077419890995, −17.838841920500467706295892666205, −17.456628963337236693140412194318, −16.35457605571014299094241685711, −15.23358612515796418329114922672, −14.53841961147372136494297718096, −13.54576789965835624723767012010, −12.52856419576086505361429838695, −11.78823648481658596463403451209, −10.574743900780437631233604543734, −10.113241886962309597046541231994, −9.540489452953245908354015418461, −8.62904280826997743158661682714, −7.03892222193483951632555205633, −5.97745630892210793627674676876, −4.85340824612142003739770127120, −4.0468421961893856949801710702, −2.673530451139832398866057273897, −2.41494871844644849979698575596, −0.02366868336198300785062455832, 1.270520291080911701746604454769, 2.72295274830472584409288080779, 4.30096213077155500302049168307, 5.408994196071196723883370614863, 6.264986067689811987060613717383, 6.72076655116290454702905502975, 8.01716291566420202878487255752, 8.63585776377663549148137272483, 9.63803768139758788860319485064, 10.57405823147092844251538885672, 12.15323878710763304385461073576, 12.878769088559132774061433570888, 13.56404847502299337040785234798, 14.0915756549898720924493787102, 15.45445789089081607764305800697, 16.34859406671731711018198901087, 17.25360149316868875147692374121, 17.350352757120737974171037838433, 18.691771705618780749279953043132, 19.32007159871644874362283676671, 20.03584682246200975909606773985, 21.75491509904638080476226224807, 22.02681175762541351625029671959, 23.39210745762458037786432754230, 23.77971959745540545867222964622

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