Properties

Label 2-4864-1.1-c1-0-116
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  + 2.32·3-s − 3.13·5-s + 0.535·7-s + 2.42·9-s + 0.425·11-s − 6.65·13-s − 7.31·15-s + 7.33·17-s − 19-s + 1.24·21-s + 5.90·23-s + 4.85·25-s − 1.33·27-s − 0.837·29-s − 3.16·31-s + 0.990·33-s − 1.68·35-s − 3.49·37-s − 15.5·39-s + 0.123·41-s − 5.39·43-s − 7.61·45-s − 2.02·47-s − 6.71·49-s + 17.0·51-s + 5.82·53-s − 1.33·55-s + ⋯
L(s)  = 1  + 1.34·3-s − 1.40·5-s + 0.202·7-s + 0.808·9-s + 0.128·11-s − 1.84·13-s − 1.88·15-s + 1.77·17-s − 0.229·19-s + 0.272·21-s + 1.23·23-s + 0.970·25-s − 0.257·27-s − 0.155·29-s − 0.569·31-s + 0.172·33-s − 0.284·35-s − 0.574·37-s − 2.48·39-s + 0.0192·41-s − 0.823·43-s − 1.13·45-s − 0.296·47-s − 0.959·49-s + 2.39·51-s + 0.800·53-s − 0.179·55-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 \( 1 - 2.32T + 3T^{2} \)
5 \( 1 + 3.13T + 5T^{2} \)
7 \( 1 - 0.535T + 7T^{2} \)
11 \( 1 - 0.425T + 11T^{2} \)
13 \( 1 + 6.65T + 13T^{2} \)
17 \( 1 - 7.33T + 17T^{2} \)
23 \( 1 - 5.90T + 23T^{2} \)
29 \( 1 + 0.837T + 29T^{2} \)
31 \( 1 + 3.16T + 31T^{2} \)
37 \( 1 + 3.49T + 37T^{2} \)
41 \( 1 - 0.123T + 41T^{2} \)
43 \( 1 + 5.39T + 43T^{2} \)
47 \( 1 + 2.02T + 47T^{2} \)
53 \( 1 - 5.82T + 53T^{2} \)
59 \( 1 + 5.56T + 59T^{2} \)
61 \( 1 - 6.99T + 61T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 - 6.99T + 73T^{2} \)
79 \( 1 - 2.07T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 - 0.801T + 97T^{2} \)
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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

−7.927820836983839126729358725021, −7.37586959636419770948640753363, −6.99196612083158255634834025500, −5.49886016677569702674706710724, −4.79174955239194762479379006591, −3.95736728867954430995635242611, −3.22614358661701648525828456404, −2.72921332885559654143229193740, −1.50717282517970140555956110305, 0, 1.50717282517970140555956110305, 2.72921332885559654143229193740, 3.22614358661701648525828456404, 3.95736728867954430995635242611, 4.79174955239194762479379006591, 5.49886016677569702674706710724, 6.99196612083158255634834025500, 7.37586959636419770948640753363, 7.927820836983839126729358725021

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