Properties

Label 1-267-267.266-r1-0-0
Degree $1$
Conductor $267$
Sign $1$
Analytic cond. $28.6931$
Root an. cond. $28.6931$
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 − 7-s − 8-s + 10-s − 11-s − 13-s + 14-s + 16-s − 17-s − 19-s − 20-s + 22-s + 23-s + 25-s + 26-s − 28-s + 29-s − 31-s − 32-s + 34-s + 35-s − 37-s + 38-s + 40-s + 41-s + ⋯
L(s)  = 1  − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 11-s − 13-s + 14-s + 16-s − 17-s − 19-s − 20-s + 22-s + 23-s + 25-s + 26-s − 28-s + 29-s − 31-s − 32-s + 34-s + 35-s − 37-s + 38-s + 40-s + 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 267 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 267 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(267\)    =    \(3 \cdot 89\)
Sign: $1$
Analytic conductor: \(28.6931\)
Root analytic conductor: \(28.6931\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{267} (266, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 267,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2759168550\)
\(L(\frac12)\) \(\approx\) \(0.2759168550\)
\(L(1)\) \(\approx\) \(0.3845246961\)
\(L(1)\) \(\approx\) \(0.3845246961\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
89 \( 1 \)
good2 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 - T \)
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 \( 1 + T \)
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

−25.82261166169045916969229778470, −24.79093279755492211153923804734, −23.84407656117716975482488352488, −23.01911797464239329923752797961, −21.83356669912771631372216146787, −20.71641855880344967948129321532, −19.62216232302816545886872494907, −19.34251029621551214453966139467, −18.3349262237306655568408668824, −17.2308263035663477445270195994, −16.28248030540042372218789585006, −15.58238268882613876793963772511, −14.83427053278911414871529356188, −12.991492268192689330923194699264, −12.312106105623701981510219806668, −11.095298230959215757686210522341, −10.35192471655556467364846542219, −9.21922729439131942872740935576, −8.2889506360555799178407556648, −7.25599721912282426408344574758, −6.518631966931917061251973962, −4.89751233144416198078169044831, −3.33864707016523830161745165922, −2.34783084378905038852647681354, −0.3488282632883660288339395669, 0.3488282632883660288339395669, 2.34783084378905038852647681354, 3.33864707016523830161745165922, 4.89751233144416198078169044831, 6.518631966931917061251973962, 7.25599721912282426408344574758, 8.2889506360555799178407556648, 9.21922729439131942872740935576, 10.35192471655556467364846542219, 11.095298230959215757686210522341, 12.312106105623701981510219806668, 12.991492268192689330923194699264, 14.83427053278911414871529356188, 15.58238268882613876793963772511, 16.28248030540042372218789585006, 17.2308263035663477445270195994, 18.3349262237306655568408668824, 19.34251029621551214453966139467, 19.62216232302816545886872494907, 20.71641855880344967948129321532, 21.83356669912771631372216146787, 23.01911797464239329923752797961, 23.84407656117716975482488352488, 24.79093279755492211153923804734, 25.82261166169045916969229778470

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