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Properties

Label 1-547-547.31-r1-0-0
Degree $1$
Conductor $547$
Sign $-0.777 - 0.629i$
Analytic cond. $58.7833$
Root an. cond. $58.7833$
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  + (−0.940 − 0.338i)2-s + (0.900 + 0.433i)3-s + (0.770 + 0.636i)4-s + (−0.509 + 0.860i)5-s + (−0.700 − 0.713i)6-s + (0.539 + 0.842i)7-s + (−0.509 − 0.860i)8-s + (0.623 + 0.781i)9-s + (0.770 − 0.636i)10-s + (0.568 − 0.822i)11-s + (0.418 + 0.908i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (−0.832 + 0.553i)15-s + (0.188 + 0.982i)16-s + (0.289 − 0.957i)17-s + ⋯
L(s)  = 1  + (−0.940 − 0.338i)2-s + (0.900 + 0.433i)3-s + (0.770 + 0.636i)4-s + (−0.509 + 0.860i)5-s + (−0.700 − 0.713i)6-s + (0.539 + 0.842i)7-s + (−0.509 − 0.860i)8-s + (0.623 + 0.781i)9-s + (0.770 − 0.636i)10-s + (0.568 − 0.822i)11-s + (0.418 + 0.908i)12-s + (−0.222 − 0.974i)13-s + (−0.222 − 0.974i)14-s + (−0.832 + 0.553i)15-s + (0.188 + 0.982i)16-s + (0.289 − 0.957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.777 - 0.629i$
Analytic conductor: \(58.7833\)
Root analytic conductor: \(58.7833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (1:\ ),\ -0.777 - 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03650375456 - 0.1030705087i\)
\(L(\frac12)\) \(\approx\) \(0.03650375456 - 0.1030705087i\)
\(L(1)\) \(\approx\) \(0.7610498724 + 0.1075793916i\)
\(L(1)\) \(\approx\) \(0.7610498724 + 0.1075793916i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.940 - 0.338i)T \)
3 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (-0.509 + 0.860i)T \)
7 \( 1 + (0.539 + 0.842i)T \)
11 \( 1 + (0.568 - 0.822i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (0.289 - 0.957i)T \)
19 \( 1 + (-0.650 - 0.759i)T \)
23 \( 1 + (-0.813 - 0.582i)T \)
29 \( 1 + (-0.985 + 0.171i)T \)
31 \( 1 + (0.596 + 0.802i)T \)
37 \( 1 + (-0.0517 + 0.998i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.978 + 0.205i)T \)
47 \( 1 + (-0.748 - 0.663i)T \)
53 \( 1 + (-0.832 + 0.553i)T \)
59 \( 1 + (-0.885 - 0.464i)T \)
61 \( 1 + (-0.851 + 0.524i)T \)
67 \( 1 + (-0.418 - 0.908i)T \)
71 \( 1 + (-0.915 - 0.402i)T \)
73 \( 1 + (-0.832 - 0.553i)T \)
79 \( 1 + (-0.978 + 0.205i)T \)
83 \( 1 + (0.354 + 0.935i)T \)
89 \( 1 + (-0.851 - 0.524i)T \)
97 \( 1 + (0.675 - 0.736i)T \)
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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

−23.8267562652385150262743812750, −23.21774859344095426074952123497, −21.33869034163587281469440366346, −20.61713124749286068274456858212, −19.99723428118154017273743151880, −19.37508183533461650108097107187, −18.645001468250349291433103424476, −17.39241280055750078060275002569, −16.99961012456261569954450404124, −16.00608081498663006482078162593, −14.89623903233204935471009177107, −14.48448011007675383954728338348, −13.32566621001084732535629424921, −12.22780223265859020524953935523, −11.51267124212177247431354685368, −10.14383180757370531913287970947, −9.43163614942459150162126371224, −8.504931625079001475211488692784, −7.82300629321010099366697097605, −7.19428378349337761319434381318, −6.11190205850520637726318589697, −4.4998656446438084717376233944, −3.75020386328800917378317554584, −1.7151013392569024519585383586, −1.602443944617833489381260205742, 0.0301277475771130241040103014, 1.733622663250947454086069851643, 2.90857807242077462391460196784, 3.24881251365105272081881181938, 4.70075731447788191445607302248, 6.25214665943653938785232581356, 7.347994484069485785607384414149, 8.220115393595002704097119981310, 8.726418118602864396142686255597, 9.78704395911297497563767627831, 10.61978338895081777154778938542, 11.42338892670643393981408020896, 12.21928629207107066848675926425, 13.54680752538291570915822035403, 14.63830062701126007894796808036, 15.27169373670810860852100943644, 15.93690834288231838926655829627, 16.9793152106124947221496267959, 18.261364872684385515074091489483, 18.58582058052549242721533336691, 19.53222775477774797087325050618, 20.102246034577451856974023380, 21.04588317890169883344793786697, 21.90834977707159962147507216515, 22.34868773837823480898502250337

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