↑

Properties

Label 1-3857-3857.3856-r0-0-0
Degree $1$
Conductor $3857$
Sign $1$
Analytic cond. $17.9118$
Root an. cond. $17.9118$
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 + 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 15-s + 16-s + 17-s + 18-s − 20-s − 22-s + 23-s − 24-s + 25-s + 26-s − 27-s + 30-s − 31-s + 32-s + 33-s + 34-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 15-s + 16-s + 17-s + 18-s − 20-s − 22-s + 23-s − 24-s + 25-s + 26-s − 27-s + 30-s − 31-s + 32-s + 33-s + 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3857 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3857 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3857\)    =    \(7 \cdot 19 \cdot 29\)
Sign: $1$
Analytic conductor: \(17.9118\)
Root analytic conductor: \(17.9118\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3857} (3856, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 3857,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.168151676\)
\(L(\frac12)\) \(\approx\) \(2.168151676\)
\(L(1)\) \(\approx\) \(1.360222869\)
\(L(1)\) \(\approx\) \(1.360222869\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
29 \( 1 \)
good2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 + T \)
23 \( 1 + T \)
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

−18.70228881681068413485526917012, −17.926045467225232943923526559035, −16.73223743578381041825668878703, −16.48966571184465435253097457833, −15.75629170740479176289899252340, −15.26398926175302005263936337569, −14.60921359124160725360430750420, −13.514561392660060941679109512405, −12.94464277344963536938909998834, −12.42329319338283289673075059017, −11.659693793731756174809115754924, −11.12792107016298351902154447210, −10.64746780927949693127903590080, −9.8705602988859433258573331147, −8.53554421054919242122889484905, −7.73302579334934834785939693485, −7.179781207884744194231917580939, −6.43815387687537554911950325494, −5.55131991081628481184645528409, −5.130289174441651037332941509933, −4.297819252594182192076001761103, −3.581821074535153169885229933708, −2.91473189664156215049488311322, −1.638078513016125677898342718887, −0.74956427908399724310479272910, 0.74956427908399724310479272910, 1.638078513016125677898342718887, 2.91473189664156215049488311322, 3.581821074535153169885229933708, 4.297819252594182192076001761103, 5.130289174441651037332941509933, 5.55131991081628481184645528409, 6.43815387687537554911950325494, 7.179781207884744194231917580939, 7.73302579334934834785939693485, 8.53554421054919242122889484905, 9.8705602988859433258573331147, 10.64746780927949693127903590080, 11.12792107016298351902154447210, 11.659693793731756174809115754924, 12.42329319338283289673075059017, 12.94464277344963536938909998834, 13.514561392660060941679109512405, 14.60921359124160725360430750420, 15.26398926175302005263936337569, 15.75629170740479176289899252340, 16.48966571184465435253097457833, 16.73223743578381041825668878703, 17.926045467225232943923526559035, 18.70228881681068413485526917012

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