Properties

Label 1-547-547.313-r0-0-0
Degree $1$
Conductor $547$
Sign $0.764 - 0.644i$
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  + (−0.819 − 0.572i)2-s + (0.623 + 0.781i)3-s + (0.343 + 0.939i)4-s + (0.709 − 0.705i)5-s + (−0.0632 − 0.997i)6-s + (−0.999 + 0.0115i)7-s + (0.256 − 0.966i)8-s + (−0.222 + 0.974i)9-s + (−0.985 + 0.171i)10-s + (0.278 − 0.960i)11-s + (−0.519 + 0.854i)12-s + (0.0747 − 0.997i)13-s + (0.826 + 0.563i)14-s + (0.993 + 0.114i)15-s + (−0.763 + 0.645i)16-s + (0.548 + 0.835i)17-s + ⋯
L(s)  = 1  + (−0.819 − 0.572i)2-s + (0.623 + 0.781i)3-s + (0.343 + 0.939i)4-s + (0.709 − 0.705i)5-s + (−0.0632 − 0.997i)6-s + (−0.999 + 0.0115i)7-s + (0.256 − 0.966i)8-s + (−0.222 + 0.974i)9-s + (−0.985 + 0.171i)10-s + (0.278 − 0.960i)11-s + (−0.519 + 0.854i)12-s + (0.0747 − 0.997i)13-s + (0.826 + 0.563i)14-s + (0.993 + 0.114i)15-s + (−0.763 + 0.645i)16-s + (0.548 + 0.835i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.764 - 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.104651030 - 0.4035429493i\)
\(L(\frac12)\) \(\approx\) \(1.104651030 - 0.4035429493i\)
\(L(1)\) \(\approx\) \(0.9369031619 - 0.1601025132i\)
\(L(1)\) \(\approx\) \(0.9369031619 - 0.1601025132i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.819 - 0.572i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (0.709 - 0.705i)T \)
7 \( 1 + (-0.999 + 0.0115i)T \)
11 \( 1 + (0.278 - 0.960i)T \)
13 \( 1 + (0.0747 - 0.997i)T \)
17 \( 1 + (0.548 + 0.835i)T \)
19 \( 1 + (-0.459 + 0.888i)T \)
23 \( 1 + (0.932 - 0.359i)T \)
29 \( 1 + (0.675 - 0.736i)T \)
31 \( 1 + (0.851 - 0.524i)T \)
37 \( 1 + (0.143 - 0.989i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.890 - 0.454i)T \)
47 \( 1 + (0.428 - 0.903i)T \)
53 \( 1 + (-0.397 - 0.917i)T \)
59 \( 1 + (0.799 + 0.600i)T \)
61 \( 1 + (-0.376 + 0.926i)T \)
67 \( 1 + (0.999 + 0.0230i)T \)
71 \( 1 + (0.586 - 0.809i)T \)
73 \( 1 + (0.993 - 0.114i)T \)
79 \( 1 + (0.0517 + 0.998i)T \)
83 \( 1 + (-0.0402 + 0.999i)T \)
89 \( 1 + (0.990 + 0.137i)T \)
97 \( 1 + (-0.667 - 0.744i)T \)
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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

−23.44027673257321384087644883339, −23.066760105046707801872725467888, −21.82942212273746728580282216948, −20.66597092002988750424414933820, −19.823292413949724073358767294353, −18.97077690817912309615722946096, −18.636347530964495859136825932340, −17.58126271478520553732707164187, −17.05257800463678381706197768445, −15.798898095522173131986386123555, −15.00327862149237024795263524224, −14.12877486371469461354346354742, −13.55463239056170843973554467437, −12.40031635377218313605778580020, −11.31105582061073700815093653021, −10.04811624132001518951689841573, −9.4487754196812614010029469975, −8.79842762770230623345817918672, −7.40331103767017836515779834878, −6.72405572963727627568212287358, −6.45204080102525834403096516998, −4.954903249634374117497681253909, −3.140821975024514203850229865583, −2.30021516104389741301151580864, −1.21875853485469093386041637659, 0.85913801978831409632287241536, 2.29431123105595501114704763692, 3.23948998925883659020260053936, 4.01106907333386510135369217382, 5.49708553400011330704305866973, 6.46881721045862671816213297941, 8.16073062050827391669632236802, 8.48874104925839639522415814055, 9.52628203175875281463174175401, 10.11096987503534673606201131210, 10.76094821899629061400564661197, 12.14855565462060726244874402685, 13.04248822433105168170911819065, 13.62490218196179097596090536244, 14.97099220182670926318364539220, 15.95332828998827535957345252606, 16.75023961841927798334776254732, 17.07071174123871425762723973335, 18.465603438919525913346979122082, 19.34990927407398446026168248338, 19.835100913228323436542556172213, 20.877443497400610241935999361741, 21.27904907382830628102510670421, 22.08489634663986222865653782977, 22.97847313883888695173273751285

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