Properties

Label 2-4864-1.1-c1-0-139
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  + 2.06·3-s + 1.45·5-s + 0.196·7-s + 1.27·9-s − 3.27·11-s − 6.72·13-s + 3.00·15-s − 3.82·17-s + 19-s + 0.406·21-s + 6.72·23-s − 2.87·25-s − 3.57·27-s + 4.26·29-s − 5.92·31-s − 6.75·33-s + 0.286·35-s − 10.1·37-s − 13.8·39-s + 7.16·41-s − 10.4·43-s + 1.84·45-s + 3.61·47-s − 6.96·49-s − 7.91·51-s + 8.52·53-s − 4.76·55-s + ⋯
L(s)  = 1  + 1.19·3-s + 0.651·5-s + 0.0743·7-s + 0.423·9-s − 0.985·11-s − 1.86·13-s + 0.777·15-s − 0.928·17-s + 0.229·19-s + 0.0887·21-s + 1.40·23-s − 0.575·25-s − 0.687·27-s + 0.792·29-s − 1.06·31-s − 1.17·33-s + 0.0484·35-s − 1.67·37-s − 2.22·39-s + 1.11·41-s − 1.58·43-s + 0.275·45-s + 0.526·47-s − 0.994·49-s − 1.10·51-s + 1.17·53-s − 0.642·55-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 \( 1 - 2.06T + 3T^{2} \)
5 \( 1 - 1.45T + 5T^{2} \)
7 \( 1 - 0.196T + 7T^{2} \)
11 \( 1 + 3.27T + 11T^{2} \)
13 \( 1 + 6.72T + 13T^{2} \)
17 \( 1 + 3.82T + 17T^{2} \)
23 \( 1 - 6.72T + 23T^{2} \)
29 \( 1 - 4.26T + 29T^{2} \)
31 \( 1 + 5.92T + 31T^{2} \)
37 \( 1 + 10.1T + 37T^{2} \)
41 \( 1 - 7.16T + 41T^{2} \)
43 \( 1 + 10.4T + 43T^{2} \)
47 \( 1 - 3.61T + 47T^{2} \)
53 \( 1 - 8.52T + 53T^{2} \)
59 \( 1 + 0.137T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 2.22T + 67T^{2} \)
71 \( 1 - 8.11T + 71T^{2} \)
73 \( 1 + 2.01T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 - 1.14T + 83T^{2} \)
89 \( 1 + 6.11T + 89T^{2} \)
97 \( 1 + 0.132T + 97T^{2} \)
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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

−7.974751379171979493450090059446, −7.23713411986579777047790146676, −6.77118020369825313939875930018, −5.39708772086965986578015060552, −5.13999291195666823776301956583, −4.08077027688608198392862999911, −2.97273382299046093186820419958, −2.51541142067134974997780798967, −1.80180423708925492745085613052, 0, 1.80180423708925492745085613052, 2.51541142067134974997780798967, 2.97273382299046093186820419958, 4.08077027688608198392862999911, 5.13999291195666823776301956583, 5.39708772086965986578015060552, 6.77118020369825313939875930018, 7.23713411986579777047790146676, 7.974751379171979493450090059446

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