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Properties

Label 1-547-547.216-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.948 - 0.316i$
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  + (−0.650 + 0.759i)2-s + (−0.900 − 0.433i)3-s + (−0.154 − 0.987i)4-s + (0.851 − 0.524i)5-s + (0.915 − 0.402i)6-s + (−0.596 + 0.802i)7-s + (0.851 + 0.524i)8-s + (0.623 + 0.781i)9-s + (−0.154 + 0.987i)10-s + (−0.748 + 0.663i)11-s + (−0.289 + 0.957i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (−0.994 + 0.103i)15-s + (−0.952 + 0.305i)16-s + (0.0517 − 0.998i)17-s + ⋯
L(s)  = 1  + (−0.650 + 0.759i)2-s + (−0.900 − 0.433i)3-s + (−0.154 − 0.987i)4-s + (0.851 − 0.524i)5-s + (0.915 − 0.402i)6-s + (−0.596 + 0.802i)7-s + (0.851 + 0.524i)8-s + (0.623 + 0.781i)9-s + (−0.154 + 0.987i)10-s + (−0.748 + 0.663i)11-s + (−0.289 + 0.957i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (−0.994 + 0.103i)15-s + (−0.952 + 0.305i)16-s + (0.0517 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.948 - 0.316i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (216, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.948 - 0.316i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001581076454 + 0.009748473350i\)
\(L(\frac12)\) \(\approx\) \(0.001581076454 + 0.009748473350i\)
\(L(1)\) \(\approx\) \(0.4677357031 + 0.07590965174i\)
\(L(1)\) \(\approx\) \(0.4677357031 + 0.07590965174i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.650 + 0.759i)T \)
3 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (0.851 - 0.524i)T \)
7 \( 1 + (-0.596 + 0.802i)T \)
11 \( 1 + (-0.748 + 0.663i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.0517 - 0.998i)T \)
19 \( 1 + (-0.832 + 0.553i)T \)
23 \( 1 + (-0.0172 + 0.999i)T \)
29 \( 1 + (-0.418 + 0.908i)T \)
31 \( 1 + (0.725 - 0.688i)T \)
37 \( 1 + (-0.792 - 0.609i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.868 - 0.495i)T \)
47 \( 1 + (-0.970 + 0.239i)T \)
53 \( 1 + (-0.994 + 0.103i)T \)
59 \( 1 + (-0.354 + 0.935i)T \)
61 \( 1 + (0.188 + 0.982i)T \)
67 \( 1 + (-0.289 + 0.957i)T \)
71 \( 1 + (0.509 - 0.860i)T \)
73 \( 1 + (-0.994 - 0.103i)T \)
79 \( 1 + (-0.868 - 0.495i)T \)
83 \( 1 + (0.120 - 0.992i)T \)
89 \( 1 + (0.188 - 0.982i)T \)
97 \( 1 + (-0.479 + 0.877i)T \)
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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

−22.70923085198538871529949004734, −21.81146662397056198851350702391, −21.35920930733513561275260223902, −20.65333122369657269739977465496, −19.29392544682189399390179938228, −18.80865209756993887535170911245, −17.78896815448336251028123960604, −17.05637556336018616683865738220, −16.61955895728426876779678399587, −15.60219727136704239928566063177, −14.216506216632866564136874308, −13.21583931447322721537071361896, −12.600141625270372663913824243, −11.31151831891326008498750627967, −10.70979765366920693635980337484, −10.11121873348071162868899239742, −9.39472197855760684081032325706, −8.19965514134187066334463134910, −6.75373971940878739991590317183, −6.36186881556912374312940215309, −4.84120166165210556777187877935, −3.864766952552287030142793788670, −2.784250946493369082108109710886, −1.52153495064125347887112988185, −0.00726642540267351351694654301, 1.49422901301353188784736465472, 2.51593757858173055276684202275, 4.79691564329152530171187016235, 5.489788877181213628533209906228, 6.02794474726606659404700694208, 7.088923849576536259074835970102, 7.96239632407358681659481515402, 9.10456557717123317826009611286, 9.945571715329738524468968492779, 10.55179011404092591870416192223, 11.90498389935746925899791509987, 12.87436399265862341672010674948, 13.38933813120125915771571328135, 14.73255208235019561630569743475, 15.74588962258694610044668064347, 16.27939459549210070974801242532, 17.28729821781317049063212379614, 17.8345300516171553845538098552, 18.45720474696152449104440928688, 19.31407450548290904988642377440, 20.41494856405684827952086209060, 21.44709274018952038902488724773, 22.54815200628463035374783577706, 22.99702607967706740825580780570, 24.02709412767712949593665997563

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