Properties

Label 2-4864-1.1-c1-0-94
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  − 1.65·3-s + 4.30·5-s − 2.73·7-s − 0.270·9-s − 1.72·11-s − 0.546·13-s − 7.11·15-s − 3.82·17-s + 19-s + 4.51·21-s + 0.546·23-s + 13.5·25-s + 5.40·27-s − 0.0738·29-s − 1.49·31-s + 2.85·33-s − 11.7·35-s + 8.56·37-s + 0.902·39-s + 1.90·41-s + 9.07·43-s − 1.16·45-s − 5.33·47-s + 0.476·49-s + 6.32·51-s − 10.1·53-s − 7.44·55-s + ⋯
L(s)  = 1  − 0.953·3-s + 1.92·5-s − 1.03·7-s − 0.0900·9-s − 0.521·11-s − 0.151·13-s − 1.83·15-s − 0.928·17-s + 0.229·19-s + 0.985·21-s + 0.113·23-s + 2.70·25-s + 1.03·27-s − 0.0137·29-s − 0.268·31-s + 0.497·33-s − 1.98·35-s + 1.40·37-s + 0.144·39-s + 0.297·41-s + 1.38·43-s − 0.173·45-s − 0.777·47-s + 0.0680·49-s + 0.885·51-s − 1.39·53-s − 1.00·55-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 \( 1 + 1.65T + 3T^{2} \)
5 \( 1 - 4.30T + 5T^{2} \)
7 \( 1 + 2.73T + 7T^{2} \)
11 \( 1 + 1.72T + 11T^{2} \)
13 \( 1 + 0.546T + 13T^{2} \)
17 \( 1 + 3.82T + 17T^{2} \)
23 \( 1 - 0.546T + 23T^{2} \)
29 \( 1 + 0.0738T + 29T^{2} \)
31 \( 1 + 1.49T + 31T^{2} \)
37 \( 1 - 8.56T + 37T^{2} \)
41 \( 1 - 1.90T + 41T^{2} \)
43 \( 1 - 9.07T + 43T^{2} \)
47 \( 1 + 5.33T + 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + 6.03T + 59T^{2} \)
61 \( 1 + 1.31T + 61T^{2} \)
67 \( 1 + 9.02T + 67T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 - 8.50T + 73T^{2} \)
79 \( 1 - 7.68T + 79T^{2} \)
83 \( 1 + 16.8T + 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 - 7.30T + 97T^{2} \)
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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

−7.82045965037293048175732969784, −6.75389526312911269630123025887, −6.31384793632612972060415243394, −5.84901273385584245717850732453, −5.21766003295515443834204270594, −4.46347377622323872318454474740, −2.97273805547147257019199507753, −2.49014690240866248215668122192, −1.31823332053005257087749146201, 0, 1.31823332053005257087749146201, 2.49014690240866248215668122192, 2.97273805547147257019199507753, 4.46347377622323872318454474740, 5.21766003295515443834204270594, 5.84901273385584245717850732453, 6.31384793632612972060415243394, 6.75389526312911269630123025887, 7.82045965037293048175732969784

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