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Properties

Label 1-547-547.529-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.474 + 0.880i$
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  + (0.895 + 0.444i)2-s + (−0.900 + 0.433i)3-s + (0.605 + 0.795i)4-s + (0.756 + 0.654i)5-s + (−0.999 − 0.0115i)6-s + (−0.958 − 0.283i)7-s + (0.188 + 0.982i)8-s + (0.623 − 0.781i)9-s + (0.386 + 0.922i)10-s + (0.692 + 0.721i)11-s + (−0.890 − 0.454i)12-s + (0.955 − 0.294i)13-s + (−0.733 − 0.680i)14-s + (−0.965 − 0.261i)15-s + (−0.267 + 0.963i)16-s + (0.924 + 0.381i)17-s + ⋯
L(s)  = 1  + (0.895 + 0.444i)2-s + (−0.900 + 0.433i)3-s + (0.605 + 0.795i)4-s + (0.756 + 0.654i)5-s + (−0.999 − 0.0115i)6-s + (−0.958 − 0.283i)7-s + (0.188 + 0.982i)8-s + (0.623 − 0.781i)9-s + (0.386 + 0.922i)10-s + (0.692 + 0.721i)11-s + (−0.890 − 0.454i)12-s + (0.955 − 0.294i)13-s + (−0.733 − 0.680i)14-s + (−0.965 − 0.261i)15-s + (−0.267 + 0.963i)16-s + (0.924 + 0.381i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.474 + 0.880i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.474 + 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.010861124 + 1.692779393i\)
\(L(\frac12)\) \(\approx\) \(1.010861124 + 1.692779393i\)
\(L(1)\) \(\approx\) \(1.230750545 + 0.8766692729i\)
\(L(1)\) \(\approx\) \(1.230750545 + 0.8766692729i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (0.895 + 0.444i)T \)
3 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (0.756 + 0.654i)T \)
7 \( 1 + (-0.958 - 0.283i)T \)
11 \( 1 + (0.692 + 0.721i)T \)
13 \( 1 + (0.955 - 0.294i)T \)
17 \( 1 + (0.924 + 0.381i)T \)
19 \( 1 + (0.586 - 0.809i)T \)
23 \( 1 + (-0.976 + 0.216i)T \)
29 \( 1 + (-0.289 + 0.957i)T \)
31 \( 1 + (0.322 + 0.946i)T \)
37 \( 1 + (-0.439 - 0.898i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.717 - 0.696i)T \)
47 \( 1 + (-0.996 + 0.0804i)T \)
53 \( 1 + (0.709 - 0.705i)T \)
59 \( 1 + (-0.919 + 0.391i)T \)
61 \( 1 + (0.211 + 0.977i)T \)
67 \( 1 + (0.838 - 0.544i)T \)
71 \( 1 + (0.0287 - 0.999i)T \)
73 \( 1 + (-0.965 + 0.261i)T \)
79 \( 1 + (0.962 - 0.272i)T \)
83 \( 1 + (-0.845 - 0.534i)T \)
89 \( 1 + (-0.952 + 0.305i)T \)
97 \( 1 + (0.548 + 0.835i)T \)
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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

−22.97326811167183173972239214354, −22.308475011216763719262023590786, −21.614962520315802063001840715602, −20.83400704982394855573722804765, −19.8741281506095691593500355030, −18.77770605319205790797956349509, −18.42285801539208663664305183893, −16.841252879912568715699206620062, −16.451166593369155068145023640610, −15.66724688571237211764273799956, −14.130129107674958519989093764518, −13.54217111638471227047638166860, −12.84795334151301775803829321768, −11.941278733604737168698000244187, −11.47053136003297507375078193982, −10.06692683275357487652658444523, −9.66795767709733441529104683437, −8.15549377612981325850466480939, −6.589049155047391808272953582597, −6.0453709886452417429828709564, −5.526181758732041870736551947433, −4.28408653893024606673588989057, −3.21404610156288161346626259752, −1.8089242636250387364444163496, −0.93364896035597248101430191112, 1.61690706535062157283743630253, 3.26585874281192541197333538598, 3.79801080692615086447965297919, 5.156537791026983323385269210505, 5.90928388340365678170033644017, 6.667539710954504398104089354821, 7.2492054559330119397215649895, 8.99143142407690693936145656724, 10.06266194787626534162057653572, 10.6984915773598551854924811615, 11.80015793174768237758869857366, 12.56614854839077674395520778905, 13.45934292821762515816501083163, 14.29643449614600949560142724490, 15.253362794168700507571074297068, 16.01896067498999096387533049590, 16.74770919760069933071488383475, 17.59788067425739778886958431360, 18.23408583268559882117965447514, 19.67386863673041790563294735279, 20.63881850715433228050290190430, 21.549753997955610623948596990083, 22.15601501727927810531897682858, 22.81188256276273505069511518474, 23.28062517322548413865260322833

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