Properties

Label 2-4800-1.1-c1-0-66
Degree $2$
Conductor $4800$
Sign $-1$
Analytic cond. $38.3281$
Root an. cond. $6.19097$
Motivic weight $1$
Arithmetic yes
Rational no
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 + 4.47·11-s − 4.47·13-s − 4.47·17-s − 2·21-s − 4·23-s + 27-s − 4·29-s + 8.94·31-s + 4.47·33-s + 4.47·37-s − 4.47·39-s + 10·41-s − 4·43-s − 8·47-s − 3·49-s − 4.47·51-s + 4.47·53-s − 13.4·59-s − 10·61-s − 2·63-s − 8·67-s − 4·69-s − 8.94·71-s − 8.94·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 0.333·9-s + 1.34·11-s − 1.24·13-s − 1.08·17-s − 0.436·21-s − 0.834·23-s + 0.192·27-s − 0.742·29-s + 1.60·31-s + 0.778·33-s + 0.735·37-s − 0.716·39-s + 1.56·41-s − 0.609·43-s − 1.16·47-s − 0.428·49-s − 0.626·51-s + 0.614·53-s − 1.74·59-s − 1.28·61-s − 0.251·63-s − 0.977·67-s − 0.481·69-s − 1.06·71-s − 1.04·73-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: no
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 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 4.47T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 - 8.94T + 31T^{2} \)
37 \( 1 - 4.47T + 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 4.47T + 53T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 + 8.94T + 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 17.8T + 97T^{2} \)
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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

−7.85829872364055419138274199540, −7.24687800066375950313695672025, −6.42922593095694567356099724039, −6.03642745594146154833359141072, −4.61632169200811206064256954699, −4.27061044370522814870291213619, −3.22887150114067356712133265076, −2.51012301705214334157245186176, −1.50912943606507894158740600361, 0, 1.50912943606507894158740600361, 2.51012301705214334157245186176, 3.22887150114067356712133265076, 4.27061044370522814870291213619, 4.61632169200811206064256954699, 6.03642745594146154833359141072, 6.42922593095694567356099724039, 7.24687800066375950313695672025, 7.85829872364055419138274199540

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