Properties

Label 2-4800-1.1-c1-0-67
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 + 27-s + 2·29-s − 4·33-s − 10·37-s − 2·39-s + 10·41-s − 4·43-s + 8·47-s − 7·49-s − 2·51-s − 10·53-s + 4·57-s − 4·59-s + 2·61-s − 12·67-s + 8·71-s − 10·73-s + 81-s − 12·83-s + 2·87-s − 6·89-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 + 0.192·27-s + 0.371·29-s − 0.696·33-s − 1.64·37-s − 0.320·39-s + 1.56·41-s − 0.609·43-s + 1.16·47-s − 49-s − 0.280·51-s − 1.37·53-s + 0.529·57-s − 0.520·59-s + 0.256·61-s − 1.46·67-s + 0.949·71-s − 1.17·73-s + 1/9·81-s − 1.31·83-s + 0.214·87-s − 0.635·89-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 \( 1 + p T^{2} \)
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 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−7.84465462281701093084420479537, −7.39546104131459972836674622338, −6.60942929553535216154216963636, −5.59929057414059862389948014289, −4.97287019204270070424993179755, −4.17881549969972779526027182436, −3.10526976657196724130917180316, −2.58465419848810939203642390782, −1.51207711394055808445468187309, 0, 1.51207711394055808445468187309, 2.58465419848810939203642390782, 3.10526976657196724130917180316, 4.17881549969972779526027182436, 4.97287019204270070424993179755, 5.59929057414059862389948014289, 6.60942929553535216154216963636, 7.39546104131459972836674622338, 7.84465462281701093084420479537

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