Properties

Label 2-80e2-1.1-c1-0-51
Degree $2$
Conductor $6400$
Sign $1$
Analytic cond. $51.1042$
Root an. cond. $7.14872$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4·7-s − 2·9-s + 3·11-s + 17-s + 7·19-s − 4·21-s + 4·23-s + 5·27-s + 8·29-s − 4·31-s − 3·33-s − 4·37-s − 3·41-s − 8·43-s + 9·49-s − 51-s + 12·53-s − 7·57-s + 8·59-s − 4·61-s − 8·63-s + 9·67-s − 4·69-s − 16·71-s + 11·73-s + 12·77-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s − 2/3·9-s + 0.904·11-s + 0.242·17-s + 1.60·19-s − 0.872·21-s + 0.834·23-s + 0.962·27-s + 1.48·29-s − 0.718·31-s − 0.522·33-s − 0.657·37-s − 0.468·41-s − 1.21·43-s + 9/7·49-s − 0.140·51-s + 1.64·53-s − 0.927·57-s + 1.04·59-s − 0.512·61-s − 1.00·63-s + 1.09·67-s − 0.481·69-s − 1.89·71-s + 1.28·73-s + 1.36·77-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: yes
Arithmetic: yes
Character: $\chi_{6400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.218805083\)
\(L(\frac12)\) \(\approx\) \(2.218805083\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.169646035539423161629243559714, −7.18990527625971341840575444335, −6.73571877809992763292503873353, −5.65041375760003099958478637730, −5.24211116159407062781829512112, −4.64458458331608682071452997285, −3.64483577428803250330853326265, −2.77618569849447454977206159618, −1.57953448661024340385182812352, −0.877726997057957550508241108235, 0.877726997057957550508241108235, 1.57953448661024340385182812352, 2.77618569849447454977206159618, 3.64483577428803250330853326265, 4.64458458331608682071452997285, 5.24211116159407062781829512112, 5.65041375760003099958478637730, 6.73571877809992763292503873353, 7.18990527625971341840575444335, 8.169646035539423161629243559714

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