Properties

Label 2-4800-1.1-c1-0-72
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 + 2·7-s + 9-s − 2·11-s − 6·13-s + 2·17-s + 2·21-s − 4·23-s + 27-s − 8·31-s − 2·33-s − 2·37-s − 6·39-s + 2·41-s + 4·43-s − 8·47-s − 3·49-s + 2·51-s − 6·53-s − 10·59-s − 2·61-s + 2·63-s + 8·67-s − 4·69-s + 12·71-s − 4·73-s − 4·77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.603·11-s − 1.66·13-s + 0.485·17-s + 0.436·21-s − 0.834·23-s + 0.192·27-s − 1.43·31-s − 0.348·33-s − 0.328·37-s − 0.960·39-s + 0.312·41-s + 0.609·43-s − 1.16·47-s − 3/7·49-s + 0.280·51-s − 0.824·53-s − 1.30·59-s − 0.256·61-s + 0.251·63-s + 0.977·67-s − 0.481·69-s + 1.42·71-s − 0.468·73-s − 0.455·77-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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−7.82425125324178640900965713696, −7.50083844850222126920204719803, −6.62229130807727725313447057117, −5.51306797780401171163763542733, −4.98557453494780998363680778315, −4.21329681476984441450446373250, −3.22247217757406603059726753652, −2.36879858490653300083733004583, −1.62764325671716933168154276530, 0, 1.62764325671716933168154276530, 2.36879858490653300083733004583, 3.22247217757406603059726753652, 4.21329681476984441450446373250, 4.98557453494780998363680778315, 5.51306797780401171163763542733, 6.62229130807727725313447057117, 7.50083844850222126920204719803, 7.82425125324178640900965713696

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