Properties

Label 2-4800-1.1-c1-0-60
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 − 4·7-s + 9-s + 4·11-s − 4·17-s − 4·21-s − 4·23-s + 27-s + 6·29-s + 4·31-s + 4·33-s − 8·37-s − 10·41-s + 4·43-s + 4·47-s + 9·49-s − 4·51-s − 12·53-s − 4·59-s − 2·61-s − 4·63-s − 4·67-s − 4·69-s + 8·73-s − 16·77-s − 12·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.970·17-s − 0.872·21-s − 0.834·23-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.696·33-s − 1.31·37-s − 1.56·41-s + 0.609·43-s + 0.583·47-s + 9/7·49-s − 0.560·51-s − 1.64·53-s − 0.520·59-s − 0.256·61-s − 0.503·63-s − 0.488·67-s − 0.481·69-s + 0.936·73-s − 1.82·77-s − 1.35·79-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: $\chi_{4800} (1, \cdot )$
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
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

−8.021654788014619511935739280648, −6.97192643207148298381861922469, −6.57460765142099130960477887405, −6.03333361016824393170176338548, −4.78799121292592452624710004322, −3.98250666298290809907579689213, −3.34720838666849936424859454753, −2.55540388846090364869553178550, −1.45236237516259834555099753652, 0, 1.45236237516259834555099753652, 2.55540388846090364869553178550, 3.34720838666849936424859454753, 3.98250666298290809907579689213, 4.78799121292592452624710004322, 6.03333361016824393170176338548, 6.57460765142099130960477887405, 6.97192643207148298381861922469, 8.021654788014619511935739280648

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