Properties

Label 2-64400-1.1-c1-0-27
Degree $2$
Conductor $64400$
Sign $-1$
Analytic cond. $514.236$
Root an. cond. $22.6767$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s − 2·9-s − 2·11-s + 13-s − 4·17-s − 2·19-s + 21-s − 23-s + 5·27-s − 3·29-s + 9·31-s + 2·33-s − 2·37-s − 39-s − 5·41-s + 4·43-s − 47-s + 49-s + 4·51-s − 6·53-s + 2·57-s + 10·61-s + 2·63-s − 14·67-s + 69-s − 3·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.603·11-s + 0.277·13-s − 0.970·17-s − 0.458·19-s + 0.218·21-s − 0.208·23-s + 0.962·27-s − 0.557·29-s + 1.61·31-s + 0.348·33-s − 0.328·37-s − 0.160·39-s − 0.780·41-s + 0.609·43-s − 0.145·47-s + 1/7·49-s + 0.560·51-s − 0.824·53-s + 0.264·57-s + 1.28·61-s + 0.251·63-s − 1.71·67-s + 0.120·69-s − 0.356·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64400\)    =    \(2^{4} \cdot 5^{2} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(514.236\)
Root analytic conductor: \(22.6767\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{64400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 64400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
23 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−14.57409547585460, −13.73741786204878, −13.59256652750310, −12.90809099163325, −12.54467226202569, −11.77718521570512, −11.56264076072272, −10.96382356325508, −10.41410253753615, −10.12147859690319, −9.363809298258867, −8.655670913144785, −8.543567238511461, −7.726372028693135, −7.189842658692268, −6.378056628598215, −6.238593029627567, −5.601987523884298, −4.893720553355854, −4.517873747045790, −3.710334099118273, −3.015226573724598, −2.479907827805497, −1.735291866602001, −0.6842360541156449, 0, 0.6842360541156449, 1.735291866602001, 2.479907827805497, 3.015226573724598, 3.710334099118273, 4.517873747045790, 4.893720553355854, 5.601987523884298, 6.238593029627567, 6.378056628598215, 7.189842658692268, 7.726372028693135, 8.543567238511461, 8.655670913144785, 9.363809298258867, 10.12147859690319, 10.41410253753615, 10.96382356325508, 11.56264076072272, 11.77718521570512, 12.54467226202569, 12.90809099163325, 13.59256652750310, 13.73741786204878, 14.57409547585460

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