Properties

Label 2-206400-1.1-c1-0-106
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 − 11-s + 13-s + 6·17-s − 8·19-s + 2·21-s − 9·23-s − 27-s + 4·29-s + 5·31-s + 33-s − 2·37-s − 39-s − 11·41-s − 43-s − 11·47-s − 3·49-s − 6·51-s − 3·53-s + 8·57-s + 5·59-s + 14·61-s − 2·63-s − 16·67-s + 9·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 1.45·17-s − 1.83·19-s + 0.436·21-s − 1.87·23-s − 0.192·27-s + 0.742·29-s + 0.898·31-s + 0.174·33-s − 0.328·37-s − 0.160·39-s − 1.71·41-s − 0.152·43-s − 1.60·47-s − 3/7·49-s − 0.840·51-s − 0.412·53-s + 1.05·57-s + 0.650·59-s + 1.79·61-s − 0.251·63-s − 1.95·67-s + 1.08·69-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: Trivial
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 \( 1 + T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−13.24446088815008, −12.76165464871359, −12.19177835208114, −12.02939570209291, −11.48396127896055, −10.90179206159862, −10.14335185498544, −10.10056141057384, −9.920566336514560, −8.950725530017982, −8.407846369867486, −8.154530010276632, −7.573988390858457, −6.873972236253825, −6.396563660212718, −6.126448770068348, −5.674302761065635, −4.847890398563283, −4.634678335309211, −3.745491295669099, −3.491894203511899, −2.792816688064691, −2.018190074732220, −1.555800374547264, −0.6033333730113693, 0, 0.6033333730113693, 1.555800374547264, 2.018190074732220, 2.792816688064691, 3.491894203511899, 3.745491295669099, 4.634678335309211, 4.847890398563283, 5.674302761065635, 6.126448770068348, 6.396563660212718, 6.873972236253825, 7.573988390858457, 8.154530010276632, 8.407846369867486, 8.950725530017982, 9.920566336514560, 10.10056141057384, 10.14335185498544, 10.90179206159862, 11.48396127896055, 12.02939570209291, 12.19177835208114, 12.76165464871359, 13.24446088815008

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