Properties

Label 2-4864-1.1-c1-0-103
Degree $2$
Conductor $4864$
Sign $-1$
Analytic cond. $38.8392$
Root an. cond. $6.23211$
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  + 2·7-s − 3·9-s − 4·11-s + 2·13-s + 2·17-s + 19-s − 2·23-s − 5·25-s + 2·29-s + 8·31-s − 10·37-s + 6·41-s + 4·43-s + 2·47-s − 3·49-s + 2·53-s − 12·59-s − 8·61-s − 6·63-s − 4·67-s − 12·71-s + 10·73-s − 8·77-s − 4·79-s + 9·81-s − 4·83-s − 6·89-s + ⋯
L(s)  = 1  + 0.755·7-s − 9-s − 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.229·19-s − 0.417·23-s − 25-s + 0.371·29-s + 1.43·31-s − 1.64·37-s + 0.937·41-s + 0.609·43-s + 0.291·47-s − 3/7·49-s + 0.274·53-s − 1.56·59-s − 1.02·61-s − 0.755·63-s − 0.488·67-s − 1.42·71-s + 1.17·73-s − 0.911·77-s − 0.450·79-s + 81-s − 0.439·83-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4864\)    =    \(2^{8} \cdot 19\)
Sign: $-1$
Analytic conductor: \(38.8392\)
Root analytic conductor: \(6.23211\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4864,\ (\ :1/2),\ -1)\)

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

−8.000555338118287366802045004402, −7.43056536951446491133299904047, −6.27312342430781451668950357662, −5.69507693605672396917445129315, −5.06378007645009501123963447883, −4.23077688008870743761779697587, −3.17610016756607359733112130734, −2.49617533529582800680760769255, −1.39427811435276876748283416588, 0, 1.39427811435276876748283416588, 2.49617533529582800680760769255, 3.17610016756607359733112130734, 4.23077688008870743761779697587, 5.06378007645009501123963447883, 5.69507693605672396917445129315, 6.27312342430781451668950357662, 7.43056536951446491133299904047, 8.000555338118287366802045004402

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