Properties

Label 1-247-247.246-r1-0-0
Degree $1$
Conductor $247$
Sign $1$
Analytic cond. $26.5438$
Root an. cond. $26.5438$
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 − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 11-s − 12-s − 14-s + 15-s + 16-s + 17-s + 18-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s − 27-s − 28-s − 29-s + 30-s + 31-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s − 11-s − 12-s − 14-s + 15-s + 16-s + 17-s + 18-s − 20-s + 21-s − 22-s + 23-s − 24-s + 25-s − 27-s − 28-s − 29-s + 30-s + 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(247\)    =    \(13 \cdot 19\)
Sign: $1$
Analytic conductor: \(26.5438\)
Root analytic conductor: \(26.5438\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{247} (246, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 247,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.845621043\)
\(L(\frac12)\) \(\approx\) \(1.845621043\)
\(L(1)\) \(\approx\) \(1.199368522\)
\(L(1)\) \(\approx\) \(1.199368522\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
19 \( 1 \)
good2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 + T \)
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

−25.75413501125129662165994721775, −24.54920558143674681976581256443, −23.6559979012906355001694475797, −23.01983438852489685109519832178, −22.579323930906148172885137019700, −21.41942784482456662492448945992, −20.57651071610694211671976577874, −19.29339851025235369738726817113, −18.701076375604666801350160185017, −17.075626470609878022497795362218, −16.1354728442884436863005448576, −15.74423103900681807020641972863, −14.67491243278944769035249909899, −13.03728123905369567610783982710, −12.699652443632535138130250827904, −11.63516206108663762331182687813, −10.85417527883220748955092824652, −9.83770878773295854437747180185, −7.82862671420420739672727451105, −7.017584292753641995114677151654, −5.94113039138656884261845705725, −4.99001466355078261249920600833, −3.89314467350453429100608041658, −2.82777682226239303258321634314, −0.768133095783889554668628125640, 0.768133095783889554668628125640, 2.82777682226239303258321634314, 3.89314467350453429100608041658, 4.99001466355078261249920600833, 5.94113039138656884261845705725, 7.017584292753641995114677151654, 7.82862671420420739672727451105, 9.83770878773295854437747180185, 10.85417527883220748955092824652, 11.63516206108663762331182687813, 12.699652443632535138130250827904, 13.03728123905369567610783982710, 14.67491243278944769035249909899, 15.74423103900681807020641972863, 16.1354728442884436863005448576, 17.075626470609878022497795362218, 18.701076375604666801350160185017, 19.29339851025235369738726817113, 20.57651071610694211671976577874, 21.41942784482456662492448945992, 22.579323930906148172885137019700, 23.01983438852489685109519832178, 23.6559979012906355001694475797, 24.54920558143674681976581256443, 25.75413501125129662165994721775

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