Properties

Label 2-4864-1.1-c1-0-121
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 + 2·5-s − 2.46·7-s − 2·9-s − 4·11-s + 2.46·13-s + 2·15-s + 5.92·17-s + 19-s − 2.46·21-s + 0.464·23-s − 25-s − 5·27-s − 4.46·29-s − 8.92·31-s − 4·33-s − 4.92·35-s + 2·37-s + 2.46·39-s + 6.92·41-s − 8.92·43-s − 4·45-s + 6.92·47-s − 0.928·49-s + 5.92·51-s − 5.53·53-s − 8·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.931·7-s − 0.666·9-s − 1.20·11-s + 0.683·13-s + 0.516·15-s + 1.43·17-s + 0.229·19-s − 0.537·21-s + 0.0967·23-s − 0.200·25-s − 0.962·27-s − 0.828·29-s − 1.60·31-s − 0.696·33-s − 0.833·35-s + 0.328·37-s + 0.394·39-s + 1.08·41-s − 1.36·43-s − 0.596·45-s + 1.01·47-s − 0.132·49-s + 0.830·51-s − 0.760·53-s − 1.07·55-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 \( 1 - 2T + 5T^{2} \)
7 \( 1 + 2.46T + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 2.46T + 13T^{2} \)
17 \( 1 - 5.92T + 17T^{2} \)
23 \( 1 - 0.464T + 23T^{2} \)
29 \( 1 + 4.46T + 29T^{2} \)
31 \( 1 + 8.92T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 6.92T + 41T^{2} \)
43 \( 1 + 8.92T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 + 5.53T + 53T^{2} \)
59 \( 1 - 3.92T + 59T^{2} \)
61 \( 1 + 4.92T + 61T^{2} \)
67 \( 1 - 7.92T + 67T^{2} \)
71 \( 1 + 14T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 + 0.928T + 97T^{2} \)
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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

−7.87684609255604018768593767162, −7.37635470433720034014778120929, −6.28536564355934383593988958156, −5.62924123225319106805395093059, −5.35746972936451176604603669929, −3.86510152142985138804017705955, −3.17904382845464435327691265716, −2.56962260051044799368333193724, −1.54972192335535122097301221981, 0, 1.54972192335535122097301221981, 2.56962260051044799368333193724, 3.17904382845464435327691265716, 3.86510152142985138804017705955, 5.35746972936451176604603669929, 5.62924123225319106805395093059, 6.28536564355934383593988958156, 7.37635470433720034014778120929, 7.87684609255604018768593767162

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