↑

Properties

Label 2-80e2-1.1-c1-0-112
Degree $2$
Conductor $6400$
Sign $-1$
Analytic cond. $51.1042$
Root an. cond. $7.14872$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 4·13-s + 2·17-s − 4·29-s − 12·37-s − 10·41-s − 7·49-s + 4·53-s + 12·61-s + 6·73-s + 9·81-s + 10·89-s + 18·97-s − 20·101-s − 20·109-s + 14·113-s − 12·117-s + ⋯
L(s)  = 1  − 9-s + 1.10·13-s + 0.485·17-s − 0.742·29-s − 1.97·37-s − 1.56·41-s − 49-s + 0.549·53-s + 1.53·61-s + 0.702·73-s + 81-s + 1.05·89-s + 1.82·97-s − 1.99·101-s − 1.91·109-s + 1.31·113-s − 1.10·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6400\)    =    \(2^{8} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(51.1042\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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

−7.78467951471813866703142211403, −6.87315856118033060742083453653, −6.27764363430354085556091551464, −5.47237371652670835325377298000, −5.02456163691776154135027092680, −3.70688844144968490550979770674, −3.41512333131181533280573297684, −2.29881549788790531514484661276, −1.32110387882551793301287621394, 0, 1.32110387882551793301287621394, 2.29881549788790531514484661276, 3.41512333131181533280573297684, 3.70688844144968490550979770674, 5.02456163691776154135027092680, 5.47237371652670835325377298000, 6.27764363430354085556091551464, 6.87315856118033060742083453653, 7.78467951471813866703142211403

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