Properties

Label 2-80e2-1.1-c1-0-38
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.16·3-s + 4.24·7-s + 7.00·9-s − 13.4·21-s − 1.41·23-s − 12.6·27-s + 8.94·29-s + 12·41-s + 3.16·43-s + 9.89·47-s + 10.9·49-s + 13.4·61-s + 29.6·63-s − 15.8·67-s + 4.47·69-s + 19.0·81-s − 9.48·83-s − 28.2·87-s − 6·89-s − 8.94·101-s − 12.7·103-s − 9.48·107-s + 13.4·109-s + ⋯
L(s)  = 1  − 1.82·3-s + 1.60·7-s + 2.33·9-s − 2.92·21-s − 0.294·23-s − 2.43·27-s + 1.66·29-s + 1.87·41-s + 0.482·43-s + 1.44·47-s + 1.57·49-s + 1.71·61-s + 3.74·63-s − 1.93·67-s + 0.538·69-s + 2.11·81-s − 1.04·83-s − 3.03·87-s − 0.635·89-s − 0.889·101-s − 1.25·103-s − 0.917·107-s + 1.28·109-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

\(L(1)\) \(\approx\) \(1.391486748\)
\(L(\frac12)\) \(\approx\) \(1.391486748\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.16T + 3T^{2} \)
7 \( 1 - 4.24T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 1.41T + 23T^{2} \)
29 \( 1 - 8.94T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 - 3.16T + 43T^{2} \)
47 \( 1 - 9.89T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 + 15.8T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 9.48T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.84698237854185036091088772531, −7.24126704166503363185719672343, −6.50953763837312410659084863284, −5.70136602064571688255490547342, −5.33788310706385739388702613838, −4.39293310433670662386798343917, −4.25985471766429859919869643486, −2.54250479292172834031722608755, −1.45030015060695382885101537357, −0.75265696496809349060463928416, 0.75265696496809349060463928416, 1.45030015060695382885101537357, 2.54250479292172834031722608755, 4.25985471766429859919869643486, 4.39293310433670662386798343917, 5.33788310706385739388702613838, 5.70136602064571688255490547342, 6.50953763837312410659084863284, 7.24126704166503363185719672343, 7.84698237854185036091088772531

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