Properties

Label 2-4864-1.1-c1-0-20
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.04·5-s − 0.418·7-s − 3·9-s + 5.27·11-s − 2.62·13-s + 3.27·17-s − 19-s − 3.46·23-s + 4.27·25-s − 2.62·29-s + 6.09·31-s + 1.27·35-s − 9.55·37-s − 4.54·41-s + 2.72·43-s + 9.13·45-s − 0.418·47-s − 6.82·49-s − 10.3·53-s − 16.0·55-s − 6.54·59-s + 3.04·61-s + 1.25·63-s + 8·65-s + 6.54·67-s + 11.3·71-s + 3.27·73-s + ⋯
L(s)  = 1  − 1.36·5-s − 0.158·7-s − 9-s + 1.59·11-s − 0.728·13-s + 0.794·17-s − 0.229·19-s − 0.722·23-s + 0.854·25-s − 0.487·29-s + 1.09·31-s + 0.215·35-s − 1.57·37-s − 0.710·41-s + 0.415·43-s + 1.36·45-s − 0.0610·47-s − 0.974·49-s − 1.42·53-s − 2.16·55-s − 0.852·59-s + 0.389·61-s + 0.158·63-s + 0.992·65-s + 0.800·67-s + 1.34·71-s + 0.383·73-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: \(0\)
Selberg data: \((2,\ 4864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9743836572\)
\(L(\frac12)\) \(\approx\) \(0.9743836572\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T^{2} \)
5 \( 1 + 3.04T + 5T^{2} \)
7 \( 1 + 0.418T + 7T^{2} \)
11 \( 1 - 5.27T + 11T^{2} \)
13 \( 1 + 2.62T + 13T^{2} \)
17 \( 1 - 3.27T + 17T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 + 2.62T + 29T^{2} \)
31 \( 1 - 6.09T + 31T^{2} \)
37 \( 1 + 9.55T + 37T^{2} \)
41 \( 1 + 4.54T + 41T^{2} \)
43 \( 1 - 2.72T + 43T^{2} \)
47 \( 1 + 0.418T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 6.54T + 59T^{2} \)
61 \( 1 - 3.04T + 61T^{2} \)
67 \( 1 - 6.54T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 3.27T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + 17.0T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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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

−8.153438381344017571522202227302, −7.72342356141037156061633249904, −6.78342774953377429427749108297, −6.27602312300742376224710039965, −5.25807034731074097981823525274, −4.46377791242291841279008734122, −3.61799567668676561165421736135, −3.20552012658871136346728065822, −1.88236790733804721724147846657, −0.53189352930242743753169850583, 0.53189352930242743753169850583, 1.88236790733804721724147846657, 3.20552012658871136346728065822, 3.61799567668676561165421736135, 4.46377791242291841279008734122, 5.25807034731074097981823525274, 6.27602312300742376224710039965, 6.78342774953377429427749108297, 7.72342356141037156061633249904, 8.153438381344017571522202227302

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