Properties

Label 2-4800-1.1-c1-0-48
Degree $2$
Conductor $4800$
Sign $-1$
Analytic cond. $38.3281$
Root an. cond. $6.19097$
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 + 9-s − 4·11-s − 2·13-s − 2·17-s + 4·19-s + 8·23-s − 27-s − 6·29-s + 8·31-s + 4·33-s + 6·37-s + 2·39-s − 6·41-s + 4·43-s − 7·49-s + 2·51-s − 2·53-s − 4·57-s − 4·59-s + 2·61-s − 4·67-s − 8·69-s + 8·71-s − 10·73-s − 8·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 49-s + 0.280·51-s − 0.274·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.488·67-s − 0.963·69-s + 0.949·71-s − 1.17·73-s − 0.900·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(38.3281\)
Root analytic conductor: \(6.19097\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4800,\ (\ :1/2),\ -1)\)

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

−7.76762359658982396998333385003, −7.24118822048955879614406549551, −6.49464747752693623226397249889, −5.59404656438966764448272320215, −5.03318175754570901585796255083, −4.41965752635916350121091913407, −3.17819102794346399133173147888, −2.51429877325105654995284787549, −1.22863180219753375386002447637, 0, 1.22863180219753375386002447637, 2.51429877325105654995284787549, 3.17819102794346399133173147888, 4.41965752635916350121091913407, 5.03318175754570901585796255083, 5.59404656438966764448272320215, 6.49464747752693623226397249889, 7.24118822048955879614406549551, 7.76762359658982396998333385003

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