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Properties

Label 1-547-547.303-r0-0-0
Degree $1$
Conductor $547$
Sign $0.868 - 0.495i$
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  + (0.915 − 0.402i)2-s + (−0.222 − 0.974i)3-s + (0.675 − 0.736i)4-s + (0.322 + 0.946i)5-s + (−0.596 − 0.802i)6-s + (0.256 + 0.966i)7-s + (0.322 − 0.946i)8-s + (−0.900 + 0.433i)9-s + (0.675 + 0.736i)10-s + (−0.748 + 0.663i)11-s + (−0.868 − 0.495i)12-s + (0.623 + 0.781i)13-s + (0.623 + 0.781i)14-s + (0.851 − 0.524i)15-s + (−0.0862 − 0.996i)16-s + (0.962 + 0.272i)17-s + ⋯
L(s)  = 1  + (0.915 − 0.402i)2-s + (−0.222 − 0.974i)3-s + (0.675 − 0.736i)4-s + (0.322 + 0.946i)5-s + (−0.596 − 0.802i)6-s + (0.256 + 0.966i)7-s + (0.322 − 0.946i)8-s + (−0.900 + 0.433i)9-s + (0.675 + 0.736i)10-s + (−0.748 + 0.663i)11-s + (−0.868 − 0.495i)12-s + (0.623 + 0.781i)13-s + (0.623 + 0.781i)14-s + (0.851 − 0.524i)15-s + (−0.0862 − 0.996i)16-s + (0.962 + 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.868 - 0.495i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.868 - 0.495i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.339986350 - 0.6211180236i\)
\(L(\frac12)\) \(\approx\) \(2.339986350 - 0.6211180236i\)
\(L(1)\) \(\approx\) \(1.730175921 - 0.4655225532i\)
\(L(1)\) \(\approx\) \(1.730175921 - 0.4655225532i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.915 - 0.402i)T \)
3 \( 1 + (-0.222 - 0.974i)T \)
5 \( 1 + (0.322 + 0.946i)T \)
7 \( 1 + (0.256 + 0.966i)T \)
11 \( 1 + (-0.748 + 0.663i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (0.962 + 0.272i)T \)
19 \( 1 + (0.509 - 0.860i)T \)
23 \( 1 + (-0.418 - 0.908i)T \)
29 \( 1 + (0.978 + 0.205i)T \)
31 \( 1 + (0.990 + 0.137i)T \)
37 \( 1 + (0.770 - 0.636i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.928 + 0.370i)T \)
47 \( 1 + (-0.970 + 0.239i)T \)
53 \( 1 + (0.851 - 0.524i)T \)
59 \( 1 + (-0.354 + 0.935i)T \)
61 \( 1 + (-0.999 - 0.0345i)T \)
67 \( 1 + (-0.868 - 0.495i)T \)
71 \( 1 + (0.725 + 0.688i)T \)
73 \( 1 + (0.851 + 0.524i)T \)
79 \( 1 + (-0.928 + 0.370i)T \)
83 \( 1 + (0.120 - 0.992i)T \)
89 \( 1 + (-0.999 + 0.0345i)T \)
97 \( 1 + (0.0517 - 0.998i)T \)
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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

−23.25073030363383156628046414828, −22.8914182475367381548083266484, −21.54423691311275770527343225118, −21.11461318825452693476168982358, −20.49985436948335973959527928170, −19.82365077732001526474771870315, −18.01313910188646940634557983079, −17.17045359288185452233553788467, −16.39936107841135068146817352483, −16.01095241434091318070695000950, −15.05272063881700383166320452152, −13.82615876461233918865281780908, −13.5992486154754922719288471996, −12.35102099819435790101732210818, −11.50417019583642822401843962435, −10.50352898350733875583641303652, −9.7681709905577226952919883961, −8.232612215309663112184740932958, −7.899926554517754857551480246710, −6.16997145050789679171469870506, −5.48719145573792332032259455242, −4.77176470732133619808833992724, −3.79765795771328252395891000984, −3.00123543423801380670896586776, −1.10994914562674918775004776435, 1.44296337797681381107511923499, 2.44506824067755586079875145426, 2.96333480378392024934348371075, 4.615755132195219545691993639491, 5.65997408244285889949420476885, 6.338790447341715460052311586332, 7.13974593997830828211936341896, 8.21767331015510910632001149440, 9.6567352084956517096559280013, 10.677597815933996931023577100381, 11.50543787535233367205993154856, 12.173357985940788153747567679567, 13.00817288704163358731064625552, 13.920597953412425361005764098931, 14.528484554351561188263881333660, 15.40168260187806129159918637910, 16.393496750204333478765270812817, 17.92948154789343829406202732783, 18.30699102834551496300424815195, 19.07788383709929524525655870960, 19.88506649503002986384793831829, 21.198067552195481996758777644206, 21.54907554894437876848770300199, 22.74992030180273134706970871629, 23.06967318906417234035729496568

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