Properties

Label 2-1840-1.1-c1-0-38
Degree $2$
Conductor $1840$
Sign $-1$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 3.23·7-s − 2·9-s − 1.23·11-s − 1.76·13-s − 15-s − 6.47·17-s − 7.70·19-s + 3.23·21-s + 23-s + 25-s − 5·27-s − 9.47·29-s + 0.236·31-s − 1.23·33-s − 3.23·35-s + 3.70·37-s − 1.76·39-s + 1.47·41-s + 9.70·43-s + 2·45-s − 0.527·47-s + 3.47·49-s − 6.47·51-s − 6·53-s + 1.23·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.22·7-s − 0.666·9-s − 0.372·11-s − 0.489·13-s − 0.258·15-s − 1.56·17-s − 1.76·19-s + 0.706·21-s + 0.208·23-s + 0.200·25-s − 0.962·27-s − 1.75·29-s + 0.0423·31-s − 0.215·33-s − 0.546·35-s + 0.609·37-s − 0.282·39-s + 0.229·41-s + 1.48·43-s + 0.298·45-s − 0.0769·47-s + 0.496·49-s − 0.906·51-s − 0.824·53-s + 0.166·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-1$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
23 \( 1 - T \)
good3 \( 1 - T + 3T^{2} \)
7 \( 1 - 3.23T + 7T^{2} \)
11 \( 1 + 1.23T + 11T^{2} \)
13 \( 1 + 1.76T + 13T^{2} \)
17 \( 1 + 6.47T + 17T^{2} \)
19 \( 1 + 7.70T + 19T^{2} \)
29 \( 1 + 9.47T + 29T^{2} \)
31 \( 1 - 0.236T + 31T^{2} \)
37 \( 1 - 3.70T + 37T^{2} \)
41 \( 1 - 1.47T + 41T^{2} \)
43 \( 1 - 9.70T + 43T^{2} \)
47 \( 1 + 0.527T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 8.94T + 59T^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 - 3.70T + 67T^{2} \)
71 \( 1 - 1.76T + 71T^{2} \)
73 \( 1 - 6.70T + 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + 2.47T + 83T^{2} \)
89 \( 1 - 7.23T + 89T^{2} \)
97 \( 1 - 16.1T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.800483037629403963701859363745, −8.076597670308677639796445279006, −7.57556698387068727437587080638, −6.54200464943715388711901528322, −5.53410208310795347432459811298, −4.58506726505908972761776961770, −3.97305926689398884035415628360, −2.59321736160339043091915744112, −1.96535384695599674898545238778, 0, 1.96535384695599674898545238778, 2.59321736160339043091915744112, 3.97305926689398884035415628360, 4.58506726505908972761776961770, 5.53410208310795347432459811298, 6.54200464943715388711901528322, 7.57556698387068727437587080638, 8.076597670308677639796445279006, 8.800483037629403963701859363745

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