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Properties

Label 1-547-547.516-r0-0-0
Degree $1$
Conductor $547$
Sign $0.277 - 0.960i$
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  + (0.940 + 0.338i)2-s + (−0.900 − 0.433i)3-s + (0.770 + 0.636i)4-s + (0.509 − 0.860i)5-s + (−0.700 − 0.713i)6-s + (−0.539 − 0.842i)7-s + (0.509 + 0.860i)8-s + (0.623 + 0.781i)9-s + (0.770 − 0.636i)10-s + (0.568 − 0.822i)11-s + (−0.418 − 0.908i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (−0.832 + 0.553i)15-s + (0.188 + 0.982i)16-s + (−0.289 + 0.957i)17-s + ⋯
L(s)  = 1  + (0.940 + 0.338i)2-s + (−0.900 − 0.433i)3-s + (0.770 + 0.636i)4-s + (0.509 − 0.860i)5-s + (−0.700 − 0.713i)6-s + (−0.539 − 0.842i)7-s + (0.509 + 0.860i)8-s + (0.623 + 0.781i)9-s + (0.770 − 0.636i)10-s + (0.568 − 0.822i)11-s + (−0.418 − 0.908i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (−0.832 + 0.553i)15-s + (0.188 + 0.982i)16-s + (−0.289 + 0.957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.277 - 0.960i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (516, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.277 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.437906643 - 1.081322408i\)
\(L(\frac12)\) \(\approx\) \(1.437906643 - 1.081322408i\)
\(L(1)\) \(\approx\) \(1.393522156 - 0.3504113594i\)
\(L(1)\) \(\approx\) \(1.393522156 - 0.3504113594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.940 + 0.338i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (0.509 - 0.860i)T \)
7 \( 1 + (-0.539 - 0.842i)T \)
11 \( 1 + (0.568 - 0.822i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (-0.289 + 0.957i)T \)
19 \( 1 + (-0.650 - 0.759i)T \)
23 \( 1 + (0.813 + 0.582i)T \)
29 \( 1 + (-0.985 + 0.171i)T \)
31 \( 1 + (-0.596 - 0.802i)T \)
37 \( 1 + (0.0517 - 0.998i)T \)
41 \( 1 + T \)
43 \( 1 + (0.978 - 0.205i)T \)
47 \( 1 + (-0.748 - 0.663i)T \)
53 \( 1 + (-0.832 + 0.553i)T \)
59 \( 1 + (0.885 + 0.464i)T \)
61 \( 1 + (0.851 - 0.524i)T \)
67 \( 1 + (-0.418 - 0.908i)T \)
71 \( 1 + (0.915 + 0.402i)T \)
73 \( 1 + (-0.832 - 0.553i)T \)
79 \( 1 + (0.978 - 0.205i)T \)
83 \( 1 + (-0.354 - 0.935i)T \)
89 \( 1 + (0.851 + 0.524i)T \)
97 \( 1 + (0.675 - 0.736i)T \)
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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

−23.103526610677671944622644628371, −22.52298118658562211927449964388, −22.16313472006706265762052622812, −21.26434711268302206363048076223, −20.66291171577420525642701787570, −19.226839461973397842441954207126, −18.681056226544101054181686135192, −17.67936923710568722196760579115, −16.593834426258333189963638466837, −15.85856922417029732987647176387, −14.79131984148685041547339189101, −14.499546381791231379622139456510, −13.08998049159737632350399516500, −12.34956588408754242275540787958, −11.54373110753435261691681703126, −10.86673212823217158992758499917, −9.76348968976063060305423582008, −9.34774027722601161202639501205, −7.01933911673521397411793364786, −6.60547476305572818061225205952, −5.74633578644536497212931975252, −4.80772143022791081219844682257, −3.84212263908458258526537209129, −2.682902299905393495071732700152, −1.69935725597966189239905590733, 0.767402603706799737765991488, 2.06283655065340029508219871310, 3.6045650181829557361729134402, 4.51847003388885408870594851545, 5.58722505053376004877352813240, 6.09688974817629994427540256730, 7.07932325096638465322312058723, 7.98967561931057892444049652420, 9.24995702448370812853576981277, 10.66474912145714388435652195670, 11.17082513126416829691010097063, 12.43807024266785038218284517421, 13.10461094583596694682139109616, 13.350581449390203633984503624160, 14.60217183175375277526883235784, 15.76704973182884400431003327643, 16.5573689336616071478241989270, 17.168791415584749784381855157235, 17.60283506992369906839270415289, 19.291137319415717424122958745688, 19.89335815248695271230018178408, 20.96631909074796939770098243515, 21.8114107631807235292661018755, 22.40875485154283535411347999188, 23.29766147949729434763456465843

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