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Properties

Label 1-547-547.535-r0-0-0
Degree $1$
Conductor $547$
Sign $0.984 + 0.173i$
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  + (−0.244 + 0.969i)2-s + (0.623 + 0.781i)3-s + (−0.880 − 0.474i)4-s + (0.300 − 0.953i)5-s + (−0.910 + 0.413i)6-s + (0.343 − 0.939i)7-s + (0.675 − 0.736i)8-s + (−0.222 + 0.974i)9-s + (0.851 + 0.524i)10-s + (−0.0402 + 0.999i)11-s + (−0.177 − 0.984i)12-s + (0.0747 − 0.997i)13-s + (0.826 + 0.563i)14-s + (0.932 − 0.359i)15-s + (0.548 + 0.835i)16-s + (−0.332 − 0.942i)17-s + ⋯
L(s)  = 1  + (−0.244 + 0.969i)2-s + (0.623 + 0.781i)3-s + (−0.880 − 0.474i)4-s + (0.300 − 0.953i)5-s + (−0.910 + 0.413i)6-s + (0.343 − 0.939i)7-s + (0.675 − 0.736i)8-s + (−0.222 + 0.974i)9-s + (0.851 + 0.524i)10-s + (−0.0402 + 0.999i)11-s + (−0.177 − 0.984i)12-s + (0.0747 − 0.997i)13-s + (0.826 + 0.563i)14-s + (0.932 − 0.359i)15-s + (0.548 + 0.835i)16-s + (−0.332 − 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.344809604 + 0.1175875642i\)
\(L(\frac12)\) \(\approx\) \(1.344809604 + 0.1175875642i\)
\(L(1)\) \(\approx\) \(1.055514360 + 0.3181763976i\)
\(L(1)\) \(\approx\) \(1.055514360 + 0.3181763976i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.244 + 0.969i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.300 - 0.953i)T \)
7 \( 1 + (0.343 - 0.939i)T \)
11 \( 1 + (-0.0402 + 0.999i)T \)
13 \( 1 + (0.0747 - 0.997i)T \)
17 \( 1 + (-0.332 - 0.942i)T \)
19 \( 1 + (-0.937 - 0.349i)T \)
23 \( 1 + (0.233 - 0.972i)T \)
29 \( 1 + (0.990 + 0.137i)T \)
31 \( 1 + (-0.418 - 0.908i)T \)
37 \( 1 + (0.895 - 0.444i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.968 - 0.250i)T \)
47 \( 1 + (0.987 - 0.160i)T \)
53 \( 1 + (-0.778 - 0.627i)T \)
59 \( 1 + (0.692 - 0.721i)T \)
61 \( 1 + (0.999 + 0.0230i)T \)
67 \( 1 + (-0.763 + 0.645i)T \)
71 \( 1 + (0.874 + 0.484i)T \)
73 \( 1 + (0.932 + 0.359i)T \)
79 \( 1 + (-0.700 - 0.713i)T \)
83 \( 1 + (0.428 + 0.903i)T \)
89 \( 1 + (-0.479 + 0.877i)T \)
97 \( 1 + (0.469 + 0.882i)T \)
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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

−23.45078423998397708997758235238, −22.17598925719186975829628943166, −21.42859567041971280096063752886, −21.13163031389314490041579254405, −19.62660444273072926840562456068, −19.12115875374251018441745540918, −18.63717946892662875244596090546, −17.84556668033111180525418303384, −17.070332242593907660333478624996, −15.498529702248705213436675042105, −14.45158869647189845412439913836, −13.961414955847437481253439831581, −13.046966571452140122596081737881, −12.11504774665775108966913008201, −11.343163826487134596555229662941, −10.54903931457011197583881069191, −9.27741336573754400015442313558, −8.67595748970417537684480070969, −7.850685867450569257487018215798, −6.58861211083986138891140880691, −5.723683551869398741887433366026, −4.05190434980175656737494231948, −3.07905375599914884581998380489, −2.260304747753753067348880692298, −1.49302767634398958768502448074, 0.75109920287140961636161328897, 2.335261323261749476642965073171, 4.09974846009843585037217689087, 4.62197260406647913435319203327, 5.37712383726235236587873972940, 6.78601084588627304019292297913, 7.83341480924718783410722650385, 8.44767736087548419617745845866, 9.424324673229133432194358372978, 10.05884499097067810902393184896, 10.92664309390672849874809719331, 12.733396521727578468518379306586, 13.34104242119870420202450461044, 14.260125352190004442789482472247, 15.00065307990198537366732871929, 15.83958699567866601414242807435, 16.58388961318764394290813341749, 17.32795626108691367277727956341, 17.975360189583693261254959891360, 19.362399174705368486212242097136, 20.38338345615129890755755387564, 20.49485197248513907287265274098, 21.80048741741997462918364262344, 22.7404063153894034907421283668, 23.47477601875708449528128050563

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