Properties

Label 2-20e2-1.1-c3-0-2
Degree $2$
Conductor $400$
Sign $1$
Analytic cond. $23.6007$
Root an. cond. $4.85806$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5·3-s − 2·7-s − 2·9-s − 39·11-s + 84·13-s − 61·17-s − 151·19-s + 10·21-s + 58·23-s + 145·27-s + 192·29-s + 18·31-s + 195·33-s − 138·37-s − 420·39-s + 229·41-s + 164·43-s + 212·47-s − 339·49-s + 305·51-s + 578·53-s + 755·57-s + 336·59-s + 858·61-s + 4·63-s + 209·67-s − 290·69-s + ⋯
L(s)  = 1  − 0.962·3-s − 0.107·7-s − 0.0740·9-s − 1.06·11-s + 1.79·13-s − 0.870·17-s − 1.82·19-s + 0.103·21-s + 0.525·23-s + 1.03·27-s + 1.22·29-s + 0.104·31-s + 1.02·33-s − 0.613·37-s − 1.72·39-s + 0.872·41-s + 0.581·43-s + 0.657·47-s − 0.988·49-s + 0.837·51-s + 1.49·53-s + 1.75·57-s + 0.741·59-s + 1.80·61-s + 0.00799·63-s + 0.381·67-s − 0.505·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(23.6007\)
Root analytic conductor: \(4.85806\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.008468385\)
\(L(\frac12)\) \(\approx\) \(1.008468385\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good3 \( 1 + 5 T + p^{3} T^{2} \)
7 \( 1 + 2 T + p^{3} T^{2} \)
11 \( 1 + 39 T + p^{3} T^{2} \)
13 \( 1 - 84 T + p^{3} T^{2} \)
17 \( 1 + 61 T + p^{3} T^{2} \)
19 \( 1 + 151 T + p^{3} T^{2} \)
23 \( 1 - 58 T + p^{3} T^{2} \)
29 \( 1 - 192 T + p^{3} T^{2} \)
31 \( 1 - 18 T + p^{3} T^{2} \)
37 \( 1 + 138 T + p^{3} T^{2} \)
41 \( 1 - 229 T + p^{3} T^{2} \)
43 \( 1 - 164 T + p^{3} T^{2} \)
47 \( 1 - 212 T + p^{3} T^{2} \)
53 \( 1 - 578 T + p^{3} T^{2} \)
59 \( 1 - 336 T + p^{3} T^{2} \)
61 \( 1 - 858 T + p^{3} T^{2} \)
67 \( 1 - 209 T + p^{3} T^{2} \)
71 \( 1 - 780 T + p^{3} T^{2} \)
73 \( 1 + 403 T + p^{3} T^{2} \)
79 \( 1 - 230 T + p^{3} T^{2} \)
83 \( 1 - 1293 T + p^{3} T^{2} \)
89 \( 1 + 1369 T + p^{3} T^{2} \)
97 \( 1 - 382 T + p^{3} T^{2} \)
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

−10.83012286836527192786361856257, −10.40070081007949510094835118567, −8.799856491221432216672537302209, −8.295762985863758344977004589960, −6.74139499385962362944463158292, −6.12506860156555018369553053888, −5.14539448262131903101312202440, −4.00597242364009796389549547487, −2.46491234231167501695306917229, −0.67265161510883313977865450518, 0.67265161510883313977865450518, 2.46491234231167501695306917229, 4.00597242364009796389549547487, 5.14539448262131903101312202440, 6.12506860156555018369553053888, 6.74139499385962362944463158292, 8.295762985863758344977004589960, 8.799856491221432216672537302209, 10.40070081007949510094835118567, 10.83012286836527192786361856257

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