Properties

Label 2-80e2-1.1-c1-0-132
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.64·3-s − 4·7-s + 4.00·9-s + 2.64·11-s − 3·17-s − 2.64·19-s − 10.5·21-s − 4·23-s + 2.64·27-s + 4·31-s + 7.00·33-s − 10.5·37-s + 5·41-s − 5.29·43-s − 8·47-s + 9·49-s − 7.93·51-s + 10.5·53-s − 7.00·57-s + 5.29·59-s − 10.5·61-s − 16.0·63-s − 7.93·67-s − 10.5·69-s − 8·71-s + 7·73-s − 10.5·77-s + ⋯
L(s)  = 1  + 1.52·3-s − 1.51·7-s + 1.33·9-s + 0.797·11-s − 0.727·17-s − 0.606·19-s − 2.30·21-s − 0.834·23-s + 0.509·27-s + 0.718·31-s + 1.21·33-s − 1.73·37-s + 0.780·41-s − 0.806·43-s − 1.16·47-s + 1.28·49-s − 1.11·51-s + 1.45·53-s − 0.927·57-s + 0.688·59-s − 1.35·61-s − 2.01·63-s − 0.969·67-s − 1.27·69-s − 0.949·71-s + 0.819·73-s − 1.20·77-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 \( 1 - 2.64T + 3T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 - 2.64T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 2.64T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 + 5.29T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 - 5.29T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + 7.93T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 7.93T + 83T^{2} \)
89 \( 1 - T + 89T^{2} \)
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

−7.80066511339698868144867013998, −6.83470490320537926686206465359, −6.60574079033666831796661054267, −5.68462424216386743750398770678, −4.39857743985684237770994546865, −3.82442928585005421839859896951, −3.18381671333882572040613513724, −2.48801286959059943482269220320, −1.61132926727736960221333292394, 0, 1.61132926727736960221333292394, 2.48801286959059943482269220320, 3.18381671333882572040613513724, 3.82442928585005421839859896951, 4.39857743985684237770994546865, 5.68462424216386743750398770678, 6.60574079033666831796661054267, 6.83470490320537926686206465359, 7.80066511339698868144867013998

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