Properties

Label 2-51600-1.1-c1-0-46
Degree $2$
Conductor $51600$
Sign $-1$
Analytic cond. $412.028$
Root an. cond. $20.2984$
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 + 9-s − 4·13-s − 6·17-s − 2·19-s − 21-s − 9·23-s − 27-s + 6·29-s − 2·31-s + 2·37-s + 4·39-s + 6·41-s − 43-s + 9·47-s − 6·49-s + 6·51-s − 6·53-s + 2·57-s + 12·59-s + 2·61-s + 63-s − 5·67-s + 9·69-s + 12·71-s − 4·73-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.10·13-s − 1.45·17-s − 0.458·19-s − 0.218·21-s − 1.87·23-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.328·37-s + 0.640·39-s + 0.937·41-s − 0.152·43-s + 1.31·47-s − 6/7·49-s + 0.840·51-s − 0.824·53-s + 0.264·57-s + 1.56·59-s + 0.256·61-s + 0.125·63-s − 0.610·67-s + 1.08·69-s + 1.42·71-s − 0.468·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51600\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 43\)
Sign: $-1$
Analytic conductor: \(412.028\)
Root analytic conductor: \(20.2984\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{51600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 51600,\ (\ :1/2),\ -1)\)

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 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
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

−14.57979846338424, −14.34716766827213, −13.76610331134312, −13.07876346249368, −12.68603730360306, −12.11154268153246, −11.69999347601947, −11.18858135017921, −10.64725377836869, −10.13292876349413, −9.672195924872420, −9.052815520209135, −8.380510619368602, −7.969246411007359, −7.252853236762221, −6.811448049397319, −6.159848659322155, −5.723579033469881, −4.959205615373227, −4.373680325266646, −4.165555300544674, −3.100262118887869, −2.181275575319050, −2.003894771875656, −0.7720637560017615, 0, 0.7720637560017615, 2.003894771875656, 2.181275575319050, 3.100262118887869, 4.165555300544674, 4.373680325266646, 4.959205615373227, 5.723579033469881, 6.159848659322155, 6.811448049397319, 7.252853236762221, 7.969246411007359, 8.380510619368602, 9.052815520209135, 9.672195924872420, 10.13292876349413, 10.64725377836869, 11.18858135017921, 11.69999347601947, 12.11154268153246, 12.68603730360306, 13.07876346249368, 13.76610331134312, 14.34716766827213, 14.57979846338424

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