Properties

Label 2-80e2-1.1-c1-0-86
Degree $2$
Conductor $6400$
Sign $-1$
Analytic cond. $51.1042$
Root an. cond. $7.14872$
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  − 2.73·3-s + 0.732·7-s + 4.46·9-s + 2·11-s + 3.46·13-s − 3.46·17-s + 0.535·19-s − 2·21-s − 6.19·23-s − 3.99·27-s + 6.92·29-s − 5.46·31-s − 5.46·33-s − 2·37-s − 9.46·39-s − 1.46·41-s − 5.26·43-s − 3.26·47-s − 6.46·49-s + 9.46·51-s + 11.4·53-s − 1.46·57-s + 7.46·59-s − 8.92·61-s + 3.26·63-s − 10.7·67-s + 16.9·69-s + ⋯
L(s)  = 1  − 1.57·3-s + 0.276·7-s + 1.48·9-s + 0.603·11-s + 0.960·13-s − 0.840·17-s + 0.122·19-s − 0.436·21-s − 1.29·23-s − 0.769·27-s + 1.28·29-s − 0.981·31-s − 0.951·33-s − 0.328·37-s − 1.51·39-s − 0.228·41-s − 0.803·43-s − 0.476·47-s − 0.923·49-s + 1.32·51-s + 1.57·53-s − 0.193·57-s + 0.971·59-s − 1.14·61-s + 0.411·63-s − 1.31·67-s + 2.03·69-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: no
Arithmetic: yes
Character: $\chi_{6400} (1, \cdot )$
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 \( 1 + 2.73T + 3T^{2} \)
7 \( 1 - 0.732T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 0.535T + 19T^{2} \)
23 \( 1 + 6.19T + 23T^{2} \)
29 \( 1 - 6.92T + 29T^{2} \)
31 \( 1 + 5.46T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 1.46T + 41T^{2} \)
43 \( 1 + 5.26T + 43T^{2} \)
47 \( 1 + 3.26T + 47T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 - 7.46T + 59T^{2} \)
61 \( 1 + 8.92T + 61T^{2} \)
67 \( 1 + 10.7T + 67T^{2} \)
71 \( 1 + 5.46T + 71T^{2} \)
73 \( 1 - 7.46T + 73T^{2} \)
79 \( 1 + 1.07T + 79T^{2} \)
83 \( 1 + 1.26T + 83T^{2} \)
89 \( 1 + 8.92T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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

−7.48441769632462767461954643604, −6.68206735959554567566606648358, −6.25590751244613118552963330408, −5.65918875960848637895850258512, −4.84384424609689517875199023981, −4.25092979716496975445499045939, −3.41870124415684562841436515501, −1.98814002947627337760997010091, −1.13804597489230440655421950718, 0, 1.13804597489230440655421950718, 1.98814002947627337760997010091, 3.41870124415684562841436515501, 4.25092979716496975445499045939, 4.84384424609689517875199023981, 5.65918875960848637895850258512, 6.25590751244613118552963330408, 6.68206735959554567566606648358, 7.48441769632462767461954643604

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