Properties

Label 1-273-273.272-r0-0-0
Degree $1$
Conductor $273$
Sign $1$
Analytic cond. $1.26780$
Root an. cond. $1.26780$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s + 11-s + 16-s + 17-s + 19-s − 20-s + 22-s − 23-s + 25-s − 29-s + 31-s + 32-s + 34-s − 37-s + 38-s − 40-s − 41-s + 43-s + 44-s − 46-s − 47-s + 50-s − 53-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s + 11-s + 16-s + 17-s + 19-s − 20-s + 22-s − 23-s + 25-s − 29-s + 31-s + 32-s + 34-s − 37-s + 38-s − 40-s − 41-s + 43-s + 44-s − 46-s − 47-s + 50-s − 53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(1.26780\)
Root analytic conductor: \(1.26780\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{273} (272, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.135333778\)
\(L(\frac12)\) \(\approx\) \(2.135333778\)
\(L(1)\) \(\approx\) \(1.762924474\)
\(L(1)\) \(\approx\) \(1.762924474\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 + T \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 + T \)
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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

−25.525033534103751769157291251418, −24.4449924891038712905471232621, −23.95754515590874946241042054553, −22.74113036818949233424926745222, −22.4689522782436516689799073454, −21.21556913702506635366823508330, −20.2685900925140278174695671392, −19.58962989535118760029728895518, −18.65361553193690841833397088630, −17.08990860966977769115236064249, −16.22725154348625848593268342217, −15.42335414359760058977392023943, −14.47862345012989325612750963641, −13.73646320577273204422228933331, −12.297792624175708439979393060424, −11.92242339352404286258000746842, −10.93994690471559864660911240276, −9.662858697434575868089805254373, −8.12266257064736596035266068051, −7.28278243141694293294808411522, −6.19982826605356080720095311379, −4.99840543013108114997026194339, −3.89714135736268736794887211873, −3.16722215495944572853226556465, −1.448945270198701683335565475259, 1.448945270198701683335565475259, 3.16722215495944572853226556465, 3.89714135736268736794887211873, 4.99840543013108114997026194339, 6.19982826605356080720095311379, 7.28278243141694293294808411522, 8.12266257064736596035266068051, 9.662858697434575868089805254373, 10.93994690471559864660911240276, 11.92242339352404286258000746842, 12.297792624175708439979393060424, 13.73646320577273204422228933331, 14.47862345012989325612750963641, 15.42335414359760058977392023943, 16.22725154348625848593268342217, 17.08990860966977769115236064249, 18.65361553193690841833397088630, 19.58962989535118760029728895518, 20.2685900925140278174695671392, 21.21556913702506635366823508330, 22.4689522782436516689799073454, 22.74113036818949233424926745222, 23.95754515590874946241042054553, 24.4449924891038712905471232621, 25.525033534103751769157291251418

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