Properties

Label 2-4864-1.1-c1-0-124
Degree $2$
Conductor $4864$
Sign $-1$
Analytic cond. $38.8392$
Root an. cond. $6.23211$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 1.73·5-s + 1.73·7-s + 9-s − 3·11-s − 3.46·15-s − 3·17-s + 19-s + 3.46·21-s + 3.46·23-s − 2.00·25-s − 4·27-s + 3.46·29-s − 6.92·31-s − 6·33-s − 2.99·35-s + 10.3·37-s − 6·41-s − 43-s − 1.73·45-s − 5.19·47-s − 4·49-s − 6·51-s + 5.19·55-s + 2·57-s − 6·59-s − 5.19·61-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.774·5-s + 0.654·7-s + 0.333·9-s − 0.904·11-s − 0.894·15-s − 0.727·17-s + 0.229·19-s + 0.755·21-s + 0.722·23-s − 0.400·25-s − 0.769·27-s + 0.643·29-s − 1.24·31-s − 1.04·33-s − 0.507·35-s + 1.70·37-s − 0.937·41-s − 0.152·43-s − 0.258·45-s − 0.757·47-s − 0.571·49-s − 0.840·51-s + 0.700·55-s + 0.264·57-s − 0.781·59-s − 0.665·61-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: no
Arithmetic: yes
Character: $\chi_{4864} (1, \cdot )$
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 \( 1 - 2T + 3T^{2} \)
5 \( 1 + 1.73T + 5T^{2} \)
7 \( 1 - 1.73T + 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 - 3.46T + 29T^{2} \)
31 \( 1 + 6.92T + 31T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + 5.19T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 5.19T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 3.46T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 4T + 97T^{2} \)
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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

−7.967601908046062758498727639263, −7.54154771182837907562951747214, −6.68402182678737925633599303039, −5.59503632385320594669237283970, −4.79398157420474018498834957177, −4.07981947445611839833999123970, −3.20456748057452351457019907730, −2.56569723035598843527809779715, −1.59507462490591482548231560399, 0, 1.59507462490591482548231560399, 2.56569723035598843527809779715, 3.20456748057452351457019907730, 4.07981947445611839833999123970, 4.79398157420474018498834957177, 5.59503632385320594669237283970, 6.68402182678737925633599303039, 7.54154771182837907562951747214, 7.967601908046062758498727639263

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