Properties

Label 2-4864-1.1-c1-0-110
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s + 7-s − 2·9-s + 4·11-s − 13-s − 2·15-s − 5·17-s + 19-s + 21-s + 9·23-s − 25-s − 5·27-s − 5·29-s − 2·31-s + 4·33-s − 2·35-s − 10·37-s − 39-s + 12·41-s + 10·43-s + 4·45-s − 12·47-s − 6·49-s − 5·51-s − 9·53-s − 8·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 0.377·7-s − 2/3·9-s + 1.20·11-s − 0.277·13-s − 0.516·15-s − 1.21·17-s + 0.229·19-s + 0.218·21-s + 1.87·23-s − 1/5·25-s − 0.962·27-s − 0.928·29-s − 0.359·31-s + 0.696·33-s − 0.338·35-s − 1.64·37-s − 0.160·39-s + 1.87·41-s + 1.52·43-s + 0.596·45-s − 1.75·47-s − 6/7·49-s − 0.700·51-s − 1.23·53-s − 1.07·55-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: yes
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 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−7.925105762829320724222848680947, −7.27016102593669119698614109672, −6.67227261307543485943205898304, −5.69049761513022482382435007500, −4.81016416757398527783630820776, −4.03773310112722838344201474726, −3.39592564013928934752432431434, −2.50538477868688370744421399200, −1.42755397424825038577276994020, 0, 1.42755397424825038577276994020, 2.50538477868688370744421399200, 3.39592564013928934752432431434, 4.03773310112722838344201474726, 4.81016416757398527783630820776, 5.69049761513022482382435007500, 6.67227261307543485943205898304, 7.27016102593669119698614109672, 7.925105762829320724222848680947

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