Properties

Label 2-4864-1.1-c1-0-115
Degree $2$
Conductor $4864$
Sign $1$
Analytic cond. $38.8392$
Root an. cond. $6.23211$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 4·5-s + 7-s + 6·9-s + 5·13-s + 12·15-s − 5·17-s + 19-s + 3·21-s − 3·23-s + 11·25-s + 9·27-s − 7·29-s − 10·31-s + 4·35-s + 2·37-s + 15·39-s + 6·41-s + 4·43-s + 24·45-s − 8·47-s − 6·49-s − 15·51-s + 9·53-s + 3·57-s − 59-s + 2·61-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.78·5-s + 0.377·7-s + 2·9-s + 1.38·13-s + 3.09·15-s − 1.21·17-s + 0.229·19-s + 0.654·21-s − 0.625·23-s + 11/5·25-s + 1.73·27-s − 1.29·29-s − 1.79·31-s + 0.676·35-s + 0.328·37-s + 2.40·39-s + 0.937·41-s + 0.609·43-s + 3.57·45-s − 1.16·47-s − 6/7·49-s − 2.10·51-s + 1.23·53-s + 0.397·57-s − 0.130·59-s + 0.256·61-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: yes
Arithmetic: yes
Character: $\chi_{4864} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.807306712\)
\(L(\frac12)\) \(\approx\) \(5.807306712\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−8.516953861454534288839268534455, −7.67400993227461590838049486915, −6.91811254047735942143181299092, −6.08169850106264326305196868557, −5.47273971444144910988344490699, −4.34064751029975278777807269437, −3.61817507134350571960281466872, −2.67838219974142762085046315525, −1.92977995463815202835161772847, −1.50372834764025438067135938320, 1.50372834764025438067135938320, 1.92977995463815202835161772847, 2.67838219974142762085046315525, 3.61817507134350571960281466872, 4.34064751029975278777807269437, 5.47273971444144910988344490699, 6.08169850106264326305196868557, 6.91811254047735942143181299092, 7.67400993227461590838049486915, 8.516953861454534288839268534455

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