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Properties

Label 1-547-547.149-r0-0-0
Degree $1$
Conductor $547$
Sign $0.278 + 0.960i$
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  + (−0.700 + 0.713i)2-s + (−0.222 + 0.974i)3-s + (−0.0172 − 0.999i)4-s + (0.725 − 0.688i)5-s + (−0.539 − 0.842i)6-s + (−0.994 + 0.103i)7-s + (0.725 + 0.688i)8-s + (−0.900 − 0.433i)9-s + (−0.0172 + 0.999i)10-s + (0.568 + 0.822i)11-s + (0.978 + 0.205i)12-s + (0.623 − 0.781i)13-s + (0.623 − 0.781i)14-s + (0.509 + 0.860i)15-s + (−0.999 + 0.0345i)16-s + (−0.868 + 0.495i)17-s + ⋯
L(s)  = 1  + (−0.700 + 0.713i)2-s + (−0.222 + 0.974i)3-s + (−0.0172 − 0.999i)4-s + (0.725 − 0.688i)5-s + (−0.539 − 0.842i)6-s + (−0.994 + 0.103i)7-s + (0.725 + 0.688i)8-s + (−0.900 − 0.433i)9-s + (−0.0172 + 0.999i)10-s + (0.568 + 0.822i)11-s + (0.978 + 0.205i)12-s + (0.623 − 0.781i)13-s + (0.623 − 0.781i)14-s + (0.509 + 0.860i)15-s + (−0.999 + 0.0345i)16-s + (−0.868 + 0.495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.278 + 0.960i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ 0.278 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7019757379 + 0.5272079803i\)
\(L(\frac12)\) \(\approx\) \(0.7019757379 + 0.5272079803i\)
\(L(1)\) \(\approx\) \(0.6736621583 + 0.3381314652i\)
\(L(1)\) \(\approx\) \(0.6736621583 + 0.3381314652i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.700 + 0.713i)T \)
3 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.725 - 0.688i)T \)
7 \( 1 + (-0.994 + 0.103i)T \)
11 \( 1 + (0.568 + 0.822i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + (-0.868 + 0.495i)T \)
19 \( 1 + (0.915 - 0.402i)T \)
23 \( 1 + (-0.985 + 0.171i)T \)
29 \( 1 + (0.386 - 0.922i)T \)
31 \( 1 + (0.256 + 0.966i)T \)
37 \( 1 + (0.962 - 0.272i)T \)
41 \( 1 + T \)
43 \( 1 + (0.449 + 0.893i)T \)
47 \( 1 + (-0.748 + 0.663i)T \)
53 \( 1 + (0.509 + 0.860i)T \)
59 \( 1 + (0.885 - 0.464i)T \)
61 \( 1 + (0.322 - 0.946i)T \)
67 \( 1 + (0.978 + 0.205i)T \)
71 \( 1 + (-0.596 - 0.802i)T \)
73 \( 1 + (0.509 - 0.860i)T \)
79 \( 1 + (0.449 + 0.893i)T \)
83 \( 1 + (-0.354 + 0.935i)T \)
89 \( 1 + (0.322 + 0.946i)T \)
97 \( 1 + (-0.289 - 0.957i)T \)
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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

−22.89188494567923692228229059658, −22.28805600025558250943804243070, −21.6949558820338203640421758694, −20.43928120121488395160188604535, −19.64474512632117904647524571737, −18.88027009469380173564030217073, −18.33947307740340424346250075332, −17.66240195391616301081952971927, −16.601732827604696248536866119798, −16.107783353330190822036072449971, −14.23108039638466120317679824179, −13.592090760550101061227542472512, −12.98518431169112699437147563133, −11.76551844057554605553619197120, −11.291274775335205291013584395537, −10.22122589803127631727479498977, −9.32776350598736998370195095052, −8.52739011718004524818097491717, −7.2467583210423090364414299438, −6.58698572717971182650876294438, −5.83587823429014021356098669128, −3.87586075694014182064499382169, −2.89058939687264595642368260152, −2.02890225666614924587763768570, −0.84619250662094945073179731297, 0.90885515384905693378636027847, 2.46235941222309513305170743480, 4.02243451264724335053683269564, 4.98873200448438133072654714301, 5.99577988831319018433183761213, 6.461740010443175665778210754390, 7.992987974163483888870978843768, 9.006515033217653736354900874238, 9.60135250245680932894982413041, 10.12040877539075568945091955488, 11.17149100536865367020307991388, 12.448209636347941786076765192818, 13.49677866248112583869861761992, 14.4097759244327406596419030235, 15.56729519789304502798164580334, 15.89934489612389682201742508600, 16.731461074065722732559033687158, 17.698418563098445252162763743, 17.92469163682739886458126923438, 19.72754760624619112088565748072, 19.927539129408367309651912279275, 20.94644245810657245427828642094, 22.1019361644971251041951058949, 22.664989718545977705470312088602, 23.52611922371065120428476468794

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