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Properties

Label 1-547-547.126-r0-0-0
Degree $1$
Conductor $547$
Sign $0.874 - 0.485i$
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  + (0.0747 − 0.997i)2-s + (0.623 − 0.781i)3-s + (−0.988 − 0.149i)4-s + (−0.733 + 0.680i)5-s + (−0.733 − 0.680i)6-s + (−0.988 + 0.149i)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 + (−0.733 + 0.680i)12-s + (0.826 − 0.563i)13-s + (0.0747 + 0.997i)14-s + (0.0747 + 0.997i)15-s + (0.955 + 0.294i)16-s + (0.955 + 0.294i)17-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)2-s + (0.623 − 0.781i)3-s + (−0.988 − 0.149i)4-s + (−0.733 + 0.680i)5-s + (−0.733 − 0.680i)6-s + (−0.988 + 0.149i)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 + (−0.733 + 0.680i)12-s + (0.826 − 0.563i)13-s + (0.0747 + 0.997i)14-s + (0.0747 + 0.997i)15-s + (0.955 + 0.294i)16-s + (0.955 + 0.294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.874 - 0.485i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.005281863 - 0.2605533128i\)
\(L(\frac12)\) \(\approx\) \(1.005281863 - 0.2605533128i\)
\(L(1)\) \(\approx\) \(0.8494745075 - 0.4166590799i\)
\(L(1)\) \(\approx\) \(0.8494745075 - 0.4166590799i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.0747 - 0.997i)T \)
3 \( 1 + (0.623 - 0.781i)T \)
5 \( 1 + (-0.733 + 0.680i)T \)
7 \( 1 + (-0.988 + 0.149i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + (0.955 + 0.294i)T \)
19 \( 1 + (0.0747 + 0.997i)T \)
23 \( 1 + (0.0747 + 0.997i)T \)
29 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (0.623 - 0.781i)T \)
37 \( 1 + (0.955 + 0.294i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.988 + 0.149i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.826 - 0.563i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.955 + 0.294i)T \)
67 \( 1 + (0.955 + 0.294i)T \)
71 \( 1 + (0.955 + 0.294i)T \)
73 \( 1 + (0.0747 - 0.997i)T \)
79 \( 1 + (0.623 + 0.781i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (0.0747 + 0.997i)T \)
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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

−23.405197722541037492800006164761, −22.86832196952228915207436283495, −21.791909582169126874292699579025, −21.075730093588449726004067164757, −20.105531751954134570903535427040, −19.09747924190434234715152641161, −18.68057566189499811696942283259, −17.018449623558680561242953402, −16.37146880003554232213319954691, −15.90755033589391858533956597947, −15.30819861161035442922648041427, −14.10416909669965182720346178355, −13.467440718797218007857947907594, −12.66467366070055965148515600480, −11.33649693561478041662601031377, −10.141475310241473201168875701388, −9.23148056859520087663748670524, −8.559907400377014924727336562794, −7.86180947435649502371461510086, −6.71014569826601384241510903060, −5.58837513532759206310391413355, −4.636227671787861895030897703431, −3.75999724791845398811367154566, −3.00757864558174666750576343241, −0.590190567039415566896106940085, 1.1491659589828431530064891132, 2.41837624811540975997502201043, 3.35383879184152624185842125363, 3.773558406499682621600797146161, 5.515345708030258755982063837115, 6.601273597724747108318778754716, 7.75985647047832430945244044332, 8.34091112592158224788615381295, 9.68889434263635508086756447941, 10.20369357846881904909526837895, 11.46293237343359873444625409646, 12.24458304716120361568196133275, 12.93727517519435227720683831011, 13.63112421846840197016940283884, 14.78341477662369174574831831915, 15.28206381038211797071968345337, 16.64725899907900055861441072830, 18.14067595795680537600471854104, 18.395041056984361913336390935864, 19.21282028907814818905089741727, 19.89715306140590376062845301487, 20.52582968051403876015182547836, 21.52321308251777169970629727154, 22.718378234991439875169580599837, 23.1710273661258141862117099583

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