Properties

Label 2-4864-1.1-c1-0-130
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  + 2.09·3-s − 3.36·5-s + 4.47·7-s + 1.39·9-s + 0.608·11-s − 1.03·13-s − 7.05·15-s − 3.06·17-s + 19-s + 9.36·21-s − 8.50·23-s + 6.34·25-s − 3.37·27-s − 7.27·29-s − 4.02·31-s + 1.27·33-s − 15.0·35-s − 4.31·37-s − 2.17·39-s + 4.15·41-s − 6.27·43-s − 4.68·45-s − 4.73·47-s + 12.9·49-s − 6.42·51-s − 6.98·53-s − 2.05·55-s + ⋯
L(s)  = 1  + 1.20·3-s − 1.50·5-s + 1.68·7-s + 0.463·9-s + 0.183·11-s − 0.288·13-s − 1.82·15-s − 0.743·17-s + 0.229·19-s + 2.04·21-s − 1.77·23-s + 1.26·25-s − 0.648·27-s − 1.35·29-s − 0.722·31-s + 0.222·33-s − 2.54·35-s − 0.708·37-s − 0.348·39-s + 0.649·41-s − 0.957·43-s − 0.698·45-s − 0.690·47-s + 1.85·49-s − 0.898·51-s − 0.958·53-s − 0.276·55-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 \( 1 - 2.09T + 3T^{2} \)
5 \( 1 + 3.36T + 5T^{2} \)
7 \( 1 - 4.47T + 7T^{2} \)
11 \( 1 - 0.608T + 11T^{2} \)
13 \( 1 + 1.03T + 13T^{2} \)
17 \( 1 + 3.06T + 17T^{2} \)
23 \( 1 + 8.50T + 23T^{2} \)
29 \( 1 + 7.27T + 29T^{2} \)
31 \( 1 + 4.02T + 31T^{2} \)
37 \( 1 + 4.31T + 37T^{2} \)
41 \( 1 - 4.15T + 41T^{2} \)
43 \( 1 + 6.27T + 43T^{2} \)
47 \( 1 + 4.73T + 47T^{2} \)
53 \( 1 + 6.98T + 53T^{2} \)
59 \( 1 + 2.64T + 59T^{2} \)
61 \( 1 - 5.11T + 61T^{2} \)
67 \( 1 - 2.62T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 - 0.913T + 79T^{2} \)
83 \( 1 + 0.887T + 83T^{2} \)
89 \( 1 - 7.61T + 89T^{2} \)
97 \( 1 + 5.93T + 97T^{2} \)
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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

−8.053295390819736421264066759032, −7.59915412272609210754211605222, −6.81045474533612778362640356898, −5.51436857343667272054156174522, −4.71776682411426978594457305887, −3.94509772289486138986031681741, −3.57271331904454812383877966050, −2.31787411629615358996354069388, −1.67354678314274588866472529600, 0, 1.67354678314274588866472529600, 2.31787411629615358996354069388, 3.57271331904454812383877966050, 3.94509772289486138986031681741, 4.71776682411426978594457305887, 5.51436857343667272054156174522, 6.81045474533612778362640356898, 7.59915412272609210754211605222, 8.053295390819736421264066759032

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