Properties

Label 2-4840-1.1-c1-0-57
Degree $2$
Conductor $4840$
Sign $-1$
Analytic cond. $38.6475$
Root an. cond. $6.21671$
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  − 2·3-s − 5-s + 9-s + 2·13-s + 2·15-s − 6·17-s − 2·23-s + 25-s + 4·27-s + 8·29-s + 4·31-s − 2·37-s − 4·39-s + 4·41-s + 4·43-s − 45-s + 2·47-s − 7·49-s + 12·51-s − 10·53-s − 8·61-s − 2·65-s − 2·67-s + 4·69-s + 8·71-s + 10·73-s − 2·75-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.554·13-s + 0.516·15-s − 1.45·17-s − 0.417·23-s + 1/5·25-s + 0.769·27-s + 1.48·29-s + 0.718·31-s − 0.328·37-s − 0.640·39-s + 0.624·41-s + 0.609·43-s − 0.149·45-s + 0.291·47-s − 49-s + 1.68·51-s − 1.37·53-s − 1.02·61-s − 0.248·65-s − 0.244·67-s + 0.481·69-s + 0.949·71-s + 1.17·73-s − 0.230·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4840\)    =    \(2^{3} \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(38.6475\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4840,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−7.973369609328016739802260600927, −6.94502565858199251934678330245, −6.38222005346185640960055390825, −5.90381201296459453545708531275, −4.79235296487685314199498668214, −4.50230140225685107342490263314, −3.40327548932625686367966986952, −2.38216406968926952108746325867, −1.06829173800301736083652189295, 0, 1.06829173800301736083652189295, 2.38216406968926952108746325867, 3.40327548932625686367966986952, 4.50230140225685107342490263314, 4.79235296487685314199498668214, 5.90381201296459453545708531275, 6.38222005346185640960055390825, 6.94502565858199251934678330245, 7.973369609328016739802260600927

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