Properties

Label 2-206400-1.1-c1-0-167
Degree $2$
Conductor $206400$
Sign $-1$
Analytic cond. $1648.11$
Root an. cond. $40.5969$
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 − 2·7-s + 9-s − 2·11-s − 2·13-s + 4·17-s − 6·19-s + 2·21-s + 6·23-s − 27-s + 10·29-s + 8·31-s + 2·33-s + 2·37-s + 2·39-s + 2·41-s + 43-s − 2·47-s − 3·49-s − 4·51-s + 10·53-s + 6·57-s + 2·59-s + 12·61-s − 2·63-s + 12·67-s − 6·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.970·17-s − 1.37·19-s + 0.436·21-s + 1.25·23-s − 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.348·33-s + 0.328·37-s + 0.320·39-s + 0.312·41-s + 0.152·43-s − 0.291·47-s − 3/7·49-s − 0.560·51-s + 1.37·53-s + 0.794·57-s + 0.260·59-s + 1.53·61-s − 0.251·63-s + 1.46·67-s − 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(206400\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 43\)
Sign: $-1$
Analytic conductor: \(1648.11\)
Root analytic conductor: \(40.5969\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{206400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 206400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
43 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.17188701774151, −12.79360427234135, −12.34828252878507, −11.86817602543513, −11.55122757253248, −10.78203198442341, −10.31728538097200, −10.21532291869545, −9.634184012595651, −9.064321971007108, −8.450362614530631, −8.111890601421689, −7.478262901270287, −6.917923609870027, −6.475288370732130, −6.195250062402587, −5.401043386717450, −5.074669037171773, −4.498154007236609, −4.000826002573458, −3.246128573745394, −2.646046719874540, −2.397393028341198, −1.242786876606136, −0.7802633924337233, 0, 0.7802633924337233, 1.242786876606136, 2.397393028341198, 2.646046719874540, 3.246128573745394, 4.000826002573458, 4.498154007236609, 5.074669037171773, 5.401043386717450, 6.195250062402587, 6.475288370732130, 6.917923609870027, 7.478262901270287, 8.111890601421689, 8.450362614530631, 9.064321971007108, 9.634184012595651, 10.21532291869545, 10.31728538097200, 10.78203198442341, 11.55122757253248, 11.86817602543513, 12.34828252878507, 12.79360427234135, 13.17188701774151

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