Properties

Label 1-1469-1469.1468-r0-0-0
Degree $1$
Conductor $1469$
Sign $1$
Analytic cond. $6.82200$
Root an. cond. $6.82200$
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 + 19-s + 20-s + 21-s + 22-s − 23-s + 24-s + 25-s − 27-s − 28-s − 29-s + 30-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 + 19-s + 20-s + 21-s + 22-s − 23-s + 24-s + 25-s − 27-s − 28-s − 29-s + 30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1469 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1469\)    =    \(13 \cdot 113\)
Sign: $1$
Analytic conductor: \(6.82200\)
Root analytic conductor: \(6.82200\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1469} (1468, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1469,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5170394344\)
\(L(\frac12)\) \(\approx\) \(0.5170394344\)
\(L(1)\) \(\approx\) \(0.4951903541\)
\(L(1)\) \(\approx\) \(0.4951903541\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
113 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
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

−20.42562581913220043767233477419, −20.052206348156599919879458579997, −18.66510894150537595374953517680, −18.419460157947996355243946927405, −17.75927654247991117174982852278, −16.97724674606299256613048734894, −16.31275536710421630920872576676, −15.83012886752914295332158988655, −14.96767810378318431935922009367, −13.53077788671782724234204179213, −13.00617238748016902275300125408, −12.18048786663524010053423432773, −11.2294898198201135602071420742, −10.56371177210453910406482429905, −9.79234182448339674935531804902, −9.490932941826818200103539518087, −8.320102031814415094356974629025, −7.16547923930360065349960592171, −6.702741901053641556569106506791, −5.71918822998511211910789200877, −5.37372029848650158666319823450, −3.80130068432054085088736478666, −2.5837320036152959949143651280, −1.82642149651246177905057791937, −0.56256642471185870279873659142, 0.56256642471185870279873659142, 1.82642149651246177905057791937, 2.5837320036152959949143651280, 3.80130068432054085088736478666, 5.37372029848650158666319823450, 5.71918822998511211910789200877, 6.702741901053641556569106506791, 7.16547923930360065349960592171, 8.320102031814415094356974629025, 9.490932941826818200103539518087, 9.79234182448339674935531804902, 10.56371177210453910406482429905, 11.2294898198201135602071420742, 12.18048786663524010053423432773, 13.00617238748016902275300125408, 13.53077788671782724234204179213, 14.96767810378318431935922009367, 15.83012886752914295332158988655, 16.31275536710421630920872576676, 16.97724674606299256613048734894, 17.75927654247991117174982852278, 18.419460157947996355243946927405, 18.66510894150537595374953517680, 20.052206348156599919879458579997, 20.42562581913220043767233477419

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