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Properties

Label 1-547-547.27-r1-0-0
Degree $1$
Conductor $547$
Sign $0.617 + 0.786i$
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  + (−0.623 + 0.781i)2-s + (0.222 + 0.974i)3-s + (−0.222 − 0.974i)4-s + (0.900 − 0.433i)5-s + (−0.900 − 0.433i)6-s + (0.222 − 0.974i)7-s + (0.900 + 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)10-s + 11-s + (0.900 − 0.433i)12-s + (0.623 + 0.781i)13-s + (0.623 + 0.781i)14-s + (0.623 + 0.781i)15-s + (−0.900 + 0.433i)16-s + (0.900 − 0.433i)17-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (0.222 + 0.974i)3-s + (−0.222 − 0.974i)4-s + (0.900 − 0.433i)5-s + (−0.900 − 0.433i)6-s + (0.222 − 0.974i)7-s + (0.900 + 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.222 + 0.974i)10-s + 11-s + (0.900 − 0.433i)12-s + (0.623 + 0.781i)13-s + (0.623 + 0.781i)14-s + (0.623 + 0.781i)15-s + (−0.900 + 0.433i)16-s + (0.900 − 0.433i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $0.617 + 0.786i$
Analytic conductor: \(58.7833\)
Root analytic conductor: \(58.7833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (1:\ ),\ 0.617 + 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.998477497 + 0.9721791311i\)
\(L(\frac12)\) \(\approx\) \(1.998477497 + 0.9721791311i\)
\(L(1)\) \(\approx\) \(1.056166202 + 0.4675157725i\)
\(L(1)\) \(\approx\) \(1.056166202 + 0.4675157725i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.623 + 0.781i)T \)
3 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (0.900 - 0.433i)T \)
7 \( 1 + (0.222 - 0.974i)T \)
11 \( 1 + T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (0.900 - 0.433i)T \)
19 \( 1 + (0.623 + 0.781i)T \)
23 \( 1 + (-0.623 - 0.781i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (0.222 + 0.974i)T \)
37 \( 1 + (0.900 - 0.433i)T \)
41 \( 1 - T \)
43 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 - T \)
61 \( 1 + (0.900 - 0.433i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 + (0.222 - 0.974i)T \)
83 \( 1 - T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 + (0.623 + 0.781i)T \)
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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

−22.715878797138972426037870153185, −22.12669373487403963736823439325, −21.32176387743298219697906790407, −20.37428965977461470929749562088, −19.6213595807170613634540351521, −18.62803868405417783902187749759, −18.27394072875800874066849566661, −17.50073475991846147462202281353, −16.79976799253601117748754785109, −15.28022316337641097662779409450, −14.30251702721156039334614677346, −13.420904686254949803467852698639, −12.73200106768126043934116897255, −11.74781245863418333509347185139, −11.19579111684546102017936795100, −9.84438757076145274608863257486, −9.180368028207274089765900399363, −8.299270162560542292214141467478, −7.40582182565978505824749042776, −6.26341928742219468683161487580, −5.47454053341919947547231387015, −3.562271140137572063949227961950, −2.73066797562124532084593488736, −1.77934073427600459391501945071, −1.00913906996838849289543651157, 0.84028511882501026498601180938, 1.864368804781305805355509144433, 3.725450045585173717347188425950, 4.57307760353386772064722044761, 5.58896992646541250538647905703, 6.411059262921315164243200131542, 7.59042484529344423659318245484, 8.646303932641620512383689767959, 9.351684771145624911110160179385, 10.04471888686919469159643740048, 10.752164377452099517792166085911, 11.92630463584302720984466927269, 13.72100732204226703153757073863, 14.04176094021254511064257851237, 14.70381608879111485263844100311, 16.02136183894515534787003250181, 16.7337946885956155609698180229, 16.92198071453519660605288467435, 18.03931335991228597263443729039, 19.0031330370496308752821708696, 20.25820699943370225047950010453, 20.486417383303644476721646648, 21.58934333379850464225323416285, 22.55252433681156115009215698777, 23.32922931889457264414121223870

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