Properties

Label 2-4864-1.1-c1-0-10
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-s − 2·5-s − 4.46·7-s − 2·9-s − 4·11-s + 4.46·13-s − 2·15-s − 7.92·17-s + 19-s − 4.46·21-s + 6.46·23-s − 25-s − 5·27-s − 2.46·29-s − 4.92·31-s − 4·33-s + 8.92·35-s − 2·37-s + 4.46·39-s − 6.92·41-s + 4.92·43-s + 4·45-s + 6.92·47-s + 12.9·49-s − 7.92·51-s + 12.4·53-s + 8·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 1.68·7-s − 0.666·9-s − 1.20·11-s + 1.23·13-s − 0.516·15-s − 1.92·17-s + 0.229·19-s − 0.974·21-s + 1.34·23-s − 0.200·25-s − 0.962·27-s − 0.457·29-s − 0.885·31-s − 0.696·33-s + 1.50·35-s − 0.328·37-s + 0.714·39-s − 1.08·41-s + 0.751·43-s + 0.596·45-s + 1.01·47-s + 1.84·49-s − 1.11·51-s + 1.71·53-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: \(0\)
Selberg data: \((2,\ 4864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5989487489\)
\(L(\frac12)\) \(\approx\) \(0.5989487489\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.46T + 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 4.46T + 13T^{2} \)
17 \( 1 + 7.92T + 17T^{2} \)
23 \( 1 - 6.46T + 23T^{2} \)
29 \( 1 + 2.46T + 29T^{2} \)
31 \( 1 + 4.92T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 - 4.92T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 + 9.92T + 59T^{2} \)
61 \( 1 + 8.92T + 61T^{2} \)
67 \( 1 + 5.92T + 67T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + 2T + 79T^{2} \)
83 \( 1 - 2.92T + 83T^{2} \)
89 \( 1 + 0.928T + 89T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
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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

−8.409844142482034624542865921103, −7.51076617999811505742478471969, −6.95558838694580589641188612609, −6.13451387595584215331300907883, −5.46671440922679715753527619258, −4.32028731814252122288459318747, −3.51893930683257661830686548440, −3.06474870964278006041252469469, −2.21907037782180396589358294394, −0.38113433795143368092123168877, 0.38113433795143368092123168877, 2.21907037782180396589358294394, 3.06474870964278006041252469469, 3.51893930683257661830686548440, 4.32028731814252122288459318747, 5.46671440922679715753527619258, 6.13451387595584215331300907883, 6.95558838694580589641188612609, 7.51076617999811505742478471969, 8.409844142482034624542865921103

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