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Properties

Label 2-4864-1.1-c1-0-122
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 4·5-s + 7-s + 6·9-s − 5·13-s + 12·15-s − 5·17-s − 19-s − 3·21-s − 3·23-s + 11·25-s − 9·27-s + 7·29-s − 10·31-s − 4·35-s − 2·37-s + 15·39-s + 6·41-s − 4·43-s − 24·45-s − 8·47-s − 6·49-s + 15·51-s − 9·53-s + 3·57-s + 59-s − 2·61-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.78·5-s + 0.377·7-s + 2·9-s − 1.38·13-s + 3.09·15-s − 1.21·17-s − 0.229·19-s − 0.654·21-s − 0.625·23-s + 11/5·25-s − 1.73·27-s + 1.29·29-s − 1.79·31-s − 0.676·35-s − 0.328·37-s + 2.40·39-s + 0.937·41-s − 0.609·43-s − 3.57·45-s − 1.16·47-s − 6/7·49-s + 2.10·51-s − 1.23·53-s + 0.397·57-s + 0.130·59-s − 0.256·61-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: \(2\)
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 \( 1 + p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−7.31356119827090209920186563162, −6.96050154009926310753542083769, −6.12771720442183076443715995433, −5.16714811516110397375730854049, −4.50399260125954826557090410675, −4.27434554400806484038081560630, −3.00732477300143878029077665745, −1.58090845027310195677042217505, 0, 0, 1.58090845027310195677042217505, 3.00732477300143878029077665745, 4.27434554400806484038081560630, 4.50399260125954826557090410675, 5.16714811516110397375730854049, 6.12771720442183076443715995433, 6.96050154009926310753542083769, 7.31356119827090209920186563162

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