Properties

Label 2-4864-1.1-c1-0-132
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  + 3-s + 3.46·5-s − 1.73·7-s − 2·9-s − 5.19·13-s + 3.46·15-s − 3·17-s − 19-s − 1.73·21-s + 1.73·23-s + 6.99·25-s − 5·27-s − 1.73·29-s − 3.46·31-s − 5.99·35-s − 5.19·39-s + 10·43-s − 6.92·45-s + 10.3·47-s − 4·49-s − 3·51-s − 5.19·53-s − 57-s − 9·59-s − 10.3·61-s + 3.46·63-s − 18·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.54·5-s − 0.654·7-s − 0.666·9-s − 1.44·13-s + 0.894·15-s − 0.727·17-s − 0.229·19-s − 0.377·21-s + 0.361·23-s + 1.39·25-s − 0.962·27-s − 0.321·29-s − 0.622·31-s − 1.01·35-s − 0.832·39-s + 1.52·43-s − 1.03·45-s + 1.51·47-s − 0.571·49-s − 0.420·51-s − 0.713·53-s − 0.132·57-s − 1.17·59-s − 1.33·61-s + 0.436·63-s − 2.23·65-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 \( 1 - T + 3T^{2} \)
5 \( 1 - 3.46T + 5T^{2} \)
7 \( 1 + 1.73T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 5.19T + 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
23 \( 1 - 1.73T + 23T^{2} \)
29 \( 1 + 1.73T + 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 5.19T + 53T^{2} \)
59 \( 1 + 9T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 13T + 67T^{2} \)
71 \( 1 - 3.46T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−7.88560844676676645445875268644, −7.17233727220419736760226530077, −6.37247181868020728973557357058, −5.77740366318043886860476785735, −5.11352737074415760587366585072, −4.15594723605491411798983175708, −2.89639130799549490579714737447, −2.55689286683143414712185358506, −1.68935833269533697548917956700, 0, 1.68935833269533697548917956700, 2.55689286683143414712185358506, 2.89639130799549490579714737447, 4.15594723605491411798983175708, 5.11352737074415760587366585072, 5.77740366318043886860476785735, 6.37247181868020728973557357058, 7.17233727220419736760226530077, 7.88560844676676645445875268644

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