Properties

Label 2-4864-1.1-c1-0-129
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  + 1.31·3-s − 0.319·5-s + 1.93·7-s − 1.25·9-s − 3.25·11-s + 4.25·13-s − 0.421·15-s + 3·17-s − 19-s + 2.55·21-s − 5.61·23-s − 4.89·25-s − 5.61·27-s − 6.55·29-s − 6.93·31-s − 4.30·33-s − 0.619·35-s − 7.57·37-s + 5.61·39-s − 4.30·41-s + 7.13·43-s + 0.402·45-s − 6.31·47-s − 3.23·49-s + 3.95·51-s − 5.31·53-s + 1.04·55-s + ⋯
L(s)  = 1  + 0.761·3-s − 0.142·5-s + 0.732·7-s − 0.419·9-s − 0.982·11-s + 1.18·13-s − 0.108·15-s + 0.727·17-s − 0.229·19-s + 0.558·21-s − 1.17·23-s − 0.979·25-s − 1.08·27-s − 1.21·29-s − 1.24·31-s − 0.748·33-s − 0.104·35-s − 1.24·37-s + 0.899·39-s − 0.671·41-s + 1.08·43-s + 0.0599·45-s − 0.921·47-s − 0.462·49-s + 0.554·51-s − 0.730·53-s + 0.140·55-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 \( 1 - 1.31T + 3T^{2} \)
5 \( 1 + 0.319T + 5T^{2} \)
7 \( 1 - 1.93T + 7T^{2} \)
11 \( 1 + 3.25T + 11T^{2} \)
13 \( 1 - 4.25T + 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
23 \( 1 + 5.61T + 23T^{2} \)
29 \( 1 + 6.55T + 29T^{2} \)
31 \( 1 + 6.93T + 31T^{2} \)
37 \( 1 + 7.57T + 37T^{2} \)
41 \( 1 + 4.30T + 41T^{2} \)
43 \( 1 - 7.13T + 43T^{2} \)
47 \( 1 + 6.31T + 47T^{2} \)
53 \( 1 + 5.31T + 53T^{2} \)
59 \( 1 + 1.74T + 59T^{2} \)
61 \( 1 - 2.19T + 61T^{2} \)
67 \( 1 - 8.68T + 67T^{2} \)
71 \( 1 - 9.15T + 71T^{2} \)
73 \( 1 - 15.9T + 73T^{2} \)
79 \( 1 + 6.93T + 79T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 + 5.69T + 89T^{2} \)
97 \( 1 - 17.1T + 97T^{2} \)
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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

−7.933625671303437000125129673343, −7.62115021668908506975788364285, −6.40956926546503215709060631410, −5.59770557009856357240932000489, −5.11650521708090271519987199436, −3.74504670475354816416291039831, −3.57038883319542126510948541507, −2.29475339358086211818247892320, −1.66608231075353477960322404983, 0, 1.66608231075353477960322404983, 2.29475339358086211818247892320, 3.57038883319542126510948541507, 3.74504670475354816416291039831, 5.11650521708090271519987199436, 5.59770557009856357240932000489, 6.40956926546503215709060631410, 7.62115021668908506975788364285, 7.933625671303437000125129673343

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