Properties

Label 2-4864-1.1-c1-0-123
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

−7.971675556599382368456341696321, −7.65372274949938642761648191113, −6.35044946705452341030747815910, −5.95693019256344265945941620264, −4.69627033409126035460272667506, −3.94308258741408995648492962752, −3.34636150144812636477948677367, −2.53875676819617014639879296368, −1.59124237865959785337354726571, 0, 1.59124237865959785337354726571, 2.53875676819617014639879296368, 3.34636150144812636477948677367, 3.94308258741408995648492962752, 4.69627033409126035460272667506, 5.95693019256344265945941620264, 6.35044946705452341030747815910, 7.65372274949938642761648191113, 7.971675556599382368456341696321

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