Properties

Label 1-19-19.18-r1-0-0
Degree $1$
Conductor $19$
Sign $1$
Analytic cond. $2.04183$
Root an. cond. $2.04183$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

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 − 13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 20-s − 21-s − 22-s + 23-s + 24-s + 25-s + 26-s − 27-s + 28-s − 29-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 − 13-s − 14-s − 15-s + 16-s + 17-s − 18-s + 20-s − 21-s − 22-s + 23-s + 24-s + 25-s + 26-s − 27-s + 28-s − 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(19\)
Sign: $1$
Analytic conductor: \(2.04183\)
Root analytic conductor: \(2.04183\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{19} (18, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 19,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8044195998\)
\(L(\frac12)\) \(\approx\) \(0.8044195998\)
\(L(1)\) \(\approx\) \(0.7207307841\)
\(L(1)\) \(\approx\) \(0.7207307841\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
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 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−40.26617802490133472247976643805, −38.82554118183282440132523450561, −37.40813650027561356755811127345, −36.37422478126958236301449933046, −34.84495686055665470774302769745, −33.87521220459731020508729447206, −32.90410771700736502136754493832, −30.126326872819792576033125367508, −29.32412200791730736572299870599, −27.963994956335836722217114002820, −27.03162527243922996956997054359, −25.15657602977425865792582365688, −24.12936469825134045135223793328, −22.01181325277693304291745500774, −20.801408834191412037687964104223, −18.76055592127636504802692087826, −17.415448573735009169608509431565, −16.86361313101393385578914242924, −14.69581220669067395382827430686, −12.176226083345964716923513454085, −10.78581062533769121248302648488, −9.383326327673246749055472537003, −7.16067082490070751048510233990, −5.47661417086705906873501833640, −1.516083753160053883697692705674, 1.516083753160053883697692705674, 5.47661417086705906873501833640, 7.16067082490070751048510233990, 9.383326327673246749055472537003, 10.78581062533769121248302648488, 12.176226083345964716923513454085, 14.69581220669067395382827430686, 16.86361313101393385578914242924, 17.415448573735009169608509431565, 18.76055592127636504802692087826, 20.801408834191412037687964104223, 22.01181325277693304291745500774, 24.12936469825134045135223793328, 25.15657602977425865792582365688, 27.03162527243922996956997054359, 27.963994956335836722217114002820, 29.32412200791730736572299870599, 30.126326872819792576033125367508, 32.90410771700736502136754493832, 33.87521220459731020508729447206, 34.84495686055665470774302769745, 36.37422478126958236301449933046, 37.40813650027561356755811127345, 38.82554118183282440132523450561, 40.26617802490133472247976643805

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