Properties

Label 2-80e2-1.1-c1-0-97
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.73·3-s + 4.73·7-s + 4.46·9-s + 3.46·11-s − 3.46·13-s + 3.46·17-s + 2·19-s + 12.9·21-s − 2.19·23-s + 3.99·27-s − 2.53·31-s + 9.46·33-s + 6·37-s − 9.46·39-s − 9.46·41-s + 0.196·43-s − 2.19·47-s + 15.3·49-s + 9.46·51-s + 10.3·53-s + 5.46·57-s + 6·59-s − 0.928·61-s + 21.1·63-s − 0.196·67-s − 6·69-s − 16.3·71-s + ⋯
L(s)  = 1  + 1.57·3-s + 1.78·7-s + 1.48·9-s + 1.04·11-s − 0.960·13-s + 0.840·17-s + 0.458·19-s + 2.82·21-s − 0.457·23-s + 0.769·27-s − 0.455·31-s + 1.64·33-s + 0.986·37-s − 1.51·39-s − 1.47·41-s + 0.0299·43-s − 0.320·47-s + 2.19·49-s + 1.32·51-s + 1.42·53-s + 0.723·57-s + 0.781·59-s − 0.118·61-s + 2.66·63-s − 0.0239·67-s − 0.722·69-s − 1.94·71-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: \(0\)
Selberg data: \((2,\ 6400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.062620197\)
\(L(\frac12)\) \(\approx\) \(5.062620197\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.73T + 7T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + 3.46T + 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 2.19T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 2.53T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 9.46T + 41T^{2} \)
43 \( 1 - 0.196T + 43T^{2} \)
47 \( 1 + 2.19T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 0.928T + 61T^{2} \)
67 \( 1 + 0.196T + 67T^{2} \)
71 \( 1 + 16.3T + 71T^{2} \)
73 \( 1 - 6.39T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 1.26T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
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

−8.159184066526061706819791352320, −7.44970655023252587998989009464, −7.07439985985922664825036190158, −5.78377769160126327336632729521, −4.98639305406565939274551113236, −4.26737423715517994140563205991, −3.61765269783335511537919938283, −2.66923009663068985978238846299, −1.88880612314076948835207481263, −1.23497447079544740800187521831, 1.23497447079544740800187521831, 1.88880612314076948835207481263, 2.66923009663068985978238846299, 3.61765269783335511537919938283, 4.26737423715517994140563205991, 4.98639305406565939274551113236, 5.78377769160126327336632729521, 7.07439985985922664825036190158, 7.44970655023252587998989009464, 8.159184066526061706819791352320

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