Properties

Label 2-684-1.1-c1-0-0
Degree $2$
Conductor $684$
Sign $1$
Analytic cond. $5.46176$
Root an. cond. $2.33704$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4.37·5-s − 2.37·7-s + 4.37·11-s + 2·13-s + 4.37·17-s + 19-s − 2.74·23-s + 14.1·25-s + 8.74·29-s + 4.74·31-s + 10.3·35-s − 6.74·37-s − 2.37·43-s + 7.62·47-s − 1.37·49-s − 8.74·53-s − 19.1·55-s − 8.37·61-s − 8.74·65-s + 13.4·67-s + 12·71-s + 0.372·73-s − 10.3·77-s + 8·79-s + 2.74·83-s − 19.1·85-s − 3.25·89-s + ⋯
L(s)  = 1  − 1.95·5-s − 0.896·7-s + 1.31·11-s + 0.554·13-s + 1.06·17-s + 0.229·19-s − 0.572·23-s + 2.82·25-s + 1.62·29-s + 0.852·31-s + 1.75·35-s − 1.10·37-s − 0.361·43-s + 1.11·47-s − 0.196·49-s − 1.20·53-s − 2.57·55-s − 1.07·61-s − 1.08·65-s + 1.64·67-s + 1.42·71-s + 0.0435·73-s − 1.18·77-s + 0.900·79-s + 0.301·83-s − 2.07·85-s − 0.345·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.003179956\)
\(L(\frac12)\) \(\approx\) \(1.003179956\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 - T \)
good5 \( 1 + 4.37T + 5T^{2} \)
7 \( 1 + 2.37T + 7T^{2} \)
11 \( 1 - 4.37T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 4.37T + 17T^{2} \)
23 \( 1 + 2.74T + 23T^{2} \)
29 \( 1 - 8.74T + 29T^{2} \)
31 \( 1 - 4.74T + 31T^{2} \)
37 \( 1 + 6.74T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 2.37T + 43T^{2} \)
47 \( 1 - 7.62T + 47T^{2} \)
53 \( 1 + 8.74T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 8.37T + 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 0.372T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 2.74T + 83T^{2} \)
89 \( 1 + 3.25T + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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

−10.56742174317230524953809301730, −9.584241768582935712165723934128, −8.585076280567286113191951860382, −7.955833152005391838770995363408, −6.94630419289265905861294352247, −6.28799776221818606249311670965, −4.73366744489186457573692788833, −3.73689180154924062249959486724, −3.24092876732629981894296988819, −0.869310282554619843558133005269, 0.869310282554619843558133005269, 3.24092876732629981894296988819, 3.73689180154924062249959486724, 4.73366744489186457573692788833, 6.28799776221818606249311670965, 6.94630419289265905861294352247, 7.955833152005391838770995363408, 8.585076280567286113191951860382, 9.584241768582935712165723934128, 10.56742174317230524953809301730

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