Properties

Label 1-547-547.312-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.361 - 0.932i$
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  + (−0.397 + 0.917i)2-s + (0.623 + 0.781i)3-s + (−0.684 − 0.729i)4-s + (−0.763 + 0.645i)5-s + (−0.965 + 0.261i)6-s + (−0.944 + 0.327i)7-s + (0.940 − 0.338i)8-s + (−0.222 + 0.974i)9-s + (−0.289 − 0.957i)10-s + (0.948 + 0.316i)11-s + (0.143 − 0.989i)12-s + (0.826 + 0.563i)13-s + (0.0747 − 0.997i)14-s + (−0.980 − 0.194i)15-s + (−0.0632 + 0.997i)16-s + (−0.910 − 0.413i)17-s + ⋯
L(s)  = 1  + (−0.397 + 0.917i)2-s + (0.623 + 0.781i)3-s + (−0.684 − 0.729i)4-s + (−0.763 + 0.645i)5-s + (−0.965 + 0.261i)6-s + (−0.944 + 0.327i)7-s + (0.940 − 0.338i)8-s + (−0.222 + 0.974i)9-s + (−0.289 − 0.957i)10-s + (0.948 + 0.316i)11-s + (0.143 − 0.989i)12-s + (0.826 + 0.563i)13-s + (0.0747 − 0.997i)14-s + (−0.980 − 0.194i)15-s + (−0.0632 + 0.997i)16-s + (−0.910 − 0.413i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.361 - 0.932i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (312, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.361 - 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3255883617 + 0.4754305817i\)
\(L(\frac12)\) \(\approx\) \(-0.3255883617 + 0.4754305817i\)
\(L(1)\) \(\approx\) \(0.3972293076 + 0.5880669896i\)
\(L(1)\) \(\approx\) \(0.3972293076 + 0.5880669896i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.397 + 0.917i)T \)
3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (-0.763 + 0.645i)T \)
7 \( 1 + (-0.944 + 0.327i)T \)
11 \( 1 + (0.948 + 0.316i)T \)
13 \( 1 + (0.826 + 0.563i)T \)
17 \( 1 + (-0.910 - 0.413i)T \)
19 \( 1 + (-0.958 + 0.283i)T \)
23 \( 1 + (-0.311 + 0.950i)T \)
29 \( 1 + (0.449 + 0.893i)T \)
31 \( 1 + (-0.952 + 0.305i)T \)
37 \( 1 + (-0.857 + 0.514i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.439 - 0.898i)T \)
47 \( 1 + (0.278 - 0.960i)T \)
53 \( 1 + (0.658 - 0.752i)T \)
59 \( 1 + (0.987 - 0.160i)T \)
61 \( 1 + (0.983 + 0.183i)T \)
67 \( 1 + (0.785 + 0.618i)T \)
71 \( 1 + (-0.614 - 0.788i)T \)
73 \( 1 + (-0.980 + 0.194i)T \)
79 \( 1 + (0.997 + 0.0689i)T \)
83 \( 1 + (-0.919 + 0.391i)T \)
89 \( 1 + (-0.650 - 0.759i)T \)
97 \( 1 + (-0.717 + 0.696i)T \)
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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

−22.86652676771374596791486857979, −21.89542679598911407800791550093, −20.66704166138511498560475450249, −20.137229186791698808824373052, −19.41301428672026445427824808978, −19.07773811471064349389749948209, −17.95641350571049342083260368215, −17.07113091433974444107702157775, −16.22993497725008819689499901031, −15.118289547043503496862165623317, −13.900853209130644398030018655226, −12.991554441894260110773074586890, −12.7077467753128777534032302083, −11.68409364600485917118499252461, −10.82316107045081197236017694754, −9.5855225309193495564908264527, −8.66046348892182102714040734542, −8.339192746345788186283682403332, −7.098385273262532231905369260846, −6.16224222378428317107259905881, −4.21972984945218122654241133284, −3.73078188351053707595406536635, −2.656248686484652485895403258175, −1.37850619472755926445492693779, −0.33524001512440867392682973518, 2.00017407326353120860093512215, 3.62040871742755181623742157288, 4.00929544290570839057294571445, 5.35602702671878454717674972465, 6.66270043590421429649721377782, 7.06783369627681973513214844494, 8.5876842463868539133495106303, 8.85278169806143477218962290644, 9.92868192229291607891098838666, 10.70596445303676446529268024048, 11.786738891460854031406804827050, 13.22915571168581940387827759529, 14.090592012462128087000855859316, 14.86995891954317435893416146684, 15.60026057680631026231689955702, 16.089731766926673549188942701774, 16.93162701130735884471522106989, 18.14099750808620093746150059362, 19.08149928013540674391520963941, 19.52294351513728454871328798296, 20.323413043859879817468006146801, 21.882474450204256212941689761167, 22.266600720775481803789934118191, 23.14975544973787728731176358025, 23.92532088579997113466345997756

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