Properties

Label 2-80e2-1.1-c1-0-41
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  − 2.23·3-s + 2.00·9-s + 2.23·11-s + 4·13-s + 3·17-s + 2.23·19-s + 8.94·23-s + 2.23·27-s + 4·29-s − 8.94·31-s − 5.00·33-s + 8·37-s − 8.94·39-s + 5·41-s + 8.94·43-s − 8.94·47-s − 7·49-s − 6.70·51-s + 4·53-s − 5.00·57-s − 8.94·59-s + 8·61-s − 6.70·67-s − 20.0·69-s − 8.94·71-s + 9·73-s − 11·81-s + ⋯
L(s)  = 1  − 1.29·3-s + 0.666·9-s + 0.674·11-s + 1.10·13-s + 0.727·17-s + 0.512·19-s + 1.86·23-s + 0.430·27-s + 0.742·29-s − 1.60·31-s − 0.870·33-s + 1.31·37-s − 1.43·39-s + 0.780·41-s + 1.36·43-s − 1.30·47-s − 49-s − 0.939·51-s + 0.549·53-s − 0.662·57-s − 1.16·59-s + 1.02·61-s − 0.819·67-s − 2.40·69-s − 1.06·71-s + 1.05·73-s − 1.22·81-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

\(L(1)\) \(\approx\) \(1.533135734\)
\(L(\frac12)\) \(\approx\) \(1.533135734\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T + 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 2.23T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 2.23T + 19T^{2} \)
23 \( 1 - 8.94T + 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 8.94T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 - 8.94T + 43T^{2} \)
47 \( 1 + 8.94T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + 8.94T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 6.70T + 67T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−7.87578198971099780593245180647, −7.18457468321643443753253307762, −6.42148271699315207010383670336, −5.95686343036384286121563316093, −5.26939069807350142130492209008, −4.60021140137269249809631041711, −3.67659336810190408368152696911, −2.89342311514597227421484031791, −1.39064664265563924031973611622, −0.78778378874472832130220495508, 0.78778378874472832130220495508, 1.39064664265563924031973611622, 2.89342311514597227421484031791, 3.67659336810190408368152696911, 4.60021140137269249809631041711, 5.26939069807350142130492209008, 5.95686343036384286121563316093, 6.42148271699315207010383670336, 7.18457468321643443753253307762, 7.87578198971099780593245180647

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