Properties

Label 2-4864-1.1-c1-0-22
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  + 1.31·5-s − 4.77·7-s − 3·9-s − 2.27·11-s + 6.09·13-s − 4.27·17-s − 19-s − 3.46·23-s − 3.27·25-s + 6.09·29-s − 2.62·31-s − 6.27·35-s − 0.837·37-s + 10.5·41-s + 10.2·43-s − 3.94·45-s − 4.77·47-s + 15.8·49-s − 10.3·53-s − 2.98·55-s + 8.54·59-s − 1.31·61-s + 14.3·63-s + 8·65-s − 8.54·67-s − 14.8·71-s − 4.27·73-s + ⋯
L(s)  = 1  + 0.587·5-s − 1.80·7-s − 9-s − 0.685·11-s + 1.68·13-s − 1.03·17-s − 0.229·19-s − 0.722·23-s − 0.654·25-s + 1.13·29-s − 0.471·31-s − 1.06·35-s − 0.137·37-s + 1.64·41-s + 1.56·43-s − 0.587·45-s − 0.696·47-s + 2.26·49-s − 1.42·53-s − 0.402·55-s + 1.11·59-s − 0.168·61-s + 1.80·63-s + 0.992·65-s − 1.04·67-s − 1.75·71-s − 0.500·73-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

\(L(1)\) \(\approx\) \(1.110032602\)
\(L(\frac12)\) \(\approx\) \(1.110032602\)
\(L(\frac{3}{2})\) 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 \( 1 + 3T^{2} \)
5 \( 1 - 1.31T + 5T^{2} \)
7 \( 1 + 4.77T + 7T^{2} \)
11 \( 1 + 2.27T + 11T^{2} \)
13 \( 1 - 6.09T + 13T^{2} \)
17 \( 1 + 4.27T + 17T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 6.09T + 29T^{2} \)
31 \( 1 + 2.62T + 31T^{2} \)
37 \( 1 + 0.837T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + 4.77T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 - 8.54T + 59T^{2} \)
61 \( 1 + 1.31T + 61T^{2} \)
67 \( 1 + 8.54T + 67T^{2} \)
71 \( 1 + 14.8T + 71T^{2} \)
73 \( 1 + 4.27T + 73T^{2} \)
79 \( 1 - 4.30T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 1.45T + 97T^{2} \)
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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

−8.431798509197689394598303712361, −7.55120418679003297127311270210, −6.50177334114577170493618171289, −6.01718346815819437691572293921, −5.82657175494581444939448157838, −4.47432731601432848201810737571, −3.57666372382721906724464424781, −2.90465325672932614713043241937, −2.11417243618199858834600937590, −0.54598775342365234803748827073, 0.54598775342365234803748827073, 2.11417243618199858834600937590, 2.90465325672932614713043241937, 3.57666372382721906724464424781, 4.47432731601432848201810737571, 5.82657175494581444939448157838, 6.01718346815819437691572293921, 6.50177334114577170493618171289, 7.55120418679003297127311270210, 8.431798509197689394598303712361

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