Properties

Label 2-20e2-1.1-c3-0-17
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  + 9·3-s + 26·7-s + 54·9-s + 59·11-s − 28·13-s − 5·17-s − 109·19-s + 234·21-s − 194·23-s + 243·27-s − 32·29-s − 10·31-s + 531·33-s + 198·37-s − 252·39-s + 117·41-s + 388·43-s − 68·47-s + 333·49-s − 45·51-s + 18·53-s − 981·57-s − 392·59-s − 710·61-s + 1.40e3·63-s − 253·67-s − 1.74e3·69-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.40·7-s + 2·9-s + 1.61·11-s − 0.597·13-s − 0.0713·17-s − 1.31·19-s + 2.43·21-s − 1.75·23-s + 1.73·27-s − 0.204·29-s − 0.0579·31-s + 2.80·33-s + 0.879·37-s − 1.03·39-s + 0.445·41-s + 1.37·43-s − 0.211·47-s + 0.970·49-s − 0.123·51-s + 0.0466·53-s − 2.27·57-s − 0.864·59-s − 1.49·61-s + 2.80·63-s − 0.461·67-s − 3.04·69-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

\(L(2)\) \(\approx\) \(4.217150837\)
\(L(\frac12)\) \(\approx\) \(4.217150837\)
\(L(\frac{5}{2})\) 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 \( 1 - p^{2} T + p^{3} T^{2} \)
7 \( 1 - 26 T + p^{3} T^{2} \)
11 \( 1 - 59 T + p^{3} T^{2} \)
13 \( 1 + 28 T + p^{3} T^{2} \)
17 \( 1 + 5 T + p^{3} T^{2} \)
19 \( 1 + 109 T + p^{3} T^{2} \)
23 \( 1 + 194 T + p^{3} T^{2} \)
29 \( 1 + 32 T + p^{3} T^{2} \)
31 \( 1 + 10 T + p^{3} T^{2} \)
37 \( 1 - 198 T + p^{3} T^{2} \)
41 \( 1 - 117 T + p^{3} T^{2} \)
43 \( 1 - 388 T + p^{3} T^{2} \)
47 \( 1 + 68 T + p^{3} T^{2} \)
53 \( 1 - 18 T + p^{3} T^{2} \)
59 \( 1 + 392 T + p^{3} T^{2} \)
61 \( 1 + 710 T + p^{3} T^{2} \)
67 \( 1 + 253 T + p^{3} T^{2} \)
71 \( 1 - 612 T + p^{3} T^{2} \)
73 \( 1 - 549 T + p^{3} T^{2} \)
79 \( 1 + 414 T + p^{3} T^{2} \)
83 \( 1 + 121 T + p^{3} T^{2} \)
89 \( 1 + 81 T + p^{3} T^{2} \)
97 \( 1 - 1502 T + p^{3} T^{2} \)
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

−10.76189716094854609410180026356, −9.608495014587047179121376136936, −8.943123469000475603149527714086, −8.127931284476167627988067387117, −7.52746210854793021305012366627, −6.26112684748082050969754083726, −4.46677087712084593790776008492, −3.92946463868303334724598949814, −2.35916375917987048982854747755, −1.55445623047435528645373196435, 1.55445623047435528645373196435, 2.35916375917987048982854747755, 3.92946463868303334724598949814, 4.46677087712084593790776008492, 6.26112684748082050969754083726, 7.52746210854793021305012366627, 8.127931284476167627988067387117, 8.943123469000475603149527714086, 9.608495014587047179121376136936, 10.76189716094854609410180026356

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