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Properties

Label 1-547-547.117-r0-0-0
Degree $1$
Conductor $547$
Sign $0.994 + 0.107i$
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  + (0.826 + 0.563i)2-s + (0.623 − 0.781i)3-s + (0.365 + 0.930i)4-s + (0.955 + 0.294i)5-s + (0.955 − 0.294i)6-s + (0.365 − 0.930i)7-s + (−0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (0.623 + 0.781i)10-s + (−0.5 − 0.866i)11-s + (0.955 + 0.294i)12-s + (0.0747 + 0.997i)13-s + (0.826 − 0.563i)14-s + (0.826 − 0.563i)15-s + (−0.733 + 0.680i)16-s + (−0.733 + 0.680i)17-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)2-s + (0.623 − 0.781i)3-s + (0.365 + 0.930i)4-s + (0.955 + 0.294i)5-s + (0.955 − 0.294i)6-s + (0.365 − 0.930i)7-s + (−0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (0.623 + 0.781i)10-s + (−0.5 − 0.866i)11-s + (0.955 + 0.294i)12-s + (0.0747 + 0.997i)13-s + (0.826 − 0.563i)14-s + (0.826 − 0.563i)15-s + (−0.733 + 0.680i)16-s + (−0.733 + 0.680i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.994 + 0.107i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.994 + 0.107i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.190376515 + 0.1715713218i\)
\(L(\frac12)\) \(\approx\) \(3.190376515 + 0.1715713218i\)
\(L(1)\) \(\approx\) \(2.253637072 + 0.1732705043i\)
\(L(1)\) \(\approx\) \(2.253637072 + 0.1732705043i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.826 + 0.563i)T \)
3 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (0.955 + 0.294i)T \)
7 \( 1 + (0.365 - 0.930i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.0747 + 0.997i)T \)
17 \( 1 + (-0.733 + 0.680i)T \)
19 \( 1 + (0.826 - 0.563i)T \)
23 \( 1 + (0.826 - 0.563i)T \)
29 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (0.623 - 0.781i)T \)
37 \( 1 + (-0.733 + 0.680i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.365 - 0.930i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.0747 + 0.997i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.733 + 0.680i)T \)
67 \( 1 + (-0.733 + 0.680i)T \)
71 \( 1 + (-0.733 + 0.680i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + (0.623 + 0.781i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (0.826 - 0.563i)T \)
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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

−22.83794270592362211772552438362, −22.517239013141798764461449331876, −21.38887007936881420004646293040, −21.05976274922637340503384812872, −20.36027175330014265656087967006, −19.57388688418723893218810274304, −18.33353334123143744403140589066, −17.69844567778795522270609240404, −16.24063293872172257672993524870, −15.38063900502948316750364763285, −14.93387931550252151426863109729, −13.883560015542754052039677513046, −13.2518097356317221154033612574, −12.36287432070693064974755276563, −11.29536849134795405962338249912, −10.278166569659088833461784015209, −9.66295002369943201487640101765, −8.88086119664684686480052751627, −7.61956690474638981100143158442, −6.10452856600170752647355711282, −5.05276841456149886735678787408, −4.89340897643932032718044292389, −3.265015418395953552439140968094, −2.5197527093108023538261084401, −1.657717446680698740055901126734, 1.4392708154942314780942385956, 2.54092706106449090204585566624, 3.44161804885907100315061330636, 4.62835078507671249517153231194, 5.79625579216885136638218524677, 6.73557194805141998726009040857, 7.22054574990556879101848362406, 8.39201047804382350529420567915, 9.12783716991760636981208798979, 10.64526170555989467324210512183, 11.45165883651249845861325387415, 12.69540871332392257605997935191, 13.582743437655193306126576523978, 13.7854786653385056380988632668, 14.55960796602946213921124901011, 15.555018905996537733623442564860, 16.79575053652030216609922850748, 17.345003496631951335260734541449, 18.26109364628745484031925408041, 19.16091172499221993453213208766, 20.39023573320116925752209111752, 20.89709156151482918407330856961, 21.758664717173912476583899107253, 22.61281914406163326882107903935, 23.82765921589526435787254726850

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