Properties

Label 2-4864-1.1-c1-0-37
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  − 2·3-s − 3.31·5-s + 3.31·7-s + 9-s + 5·11-s + 6.63·15-s + 5·17-s + 19-s − 6.63·21-s + 6.63·23-s + 6·25-s + 4·27-s − 6.63·29-s − 10·33-s − 11·35-s + 6.63·37-s + 6·41-s − 43-s − 3.31·45-s − 9.94·47-s + 4·49-s − 10·51-s + 13.2·53-s − 16.5·55-s − 2·57-s − 6·59-s − 9.94·61-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.48·5-s + 1.25·7-s + 0.333·9-s + 1.50·11-s + 1.71·15-s + 1.21·17-s + 0.229·19-s − 1.44·21-s + 1.38·23-s + 1.20·25-s + 0.769·27-s − 1.23·29-s − 1.74·33-s − 1.85·35-s + 1.09·37-s + 0.937·41-s − 0.152·43-s − 0.494·45-s − 1.45·47-s + 0.571·49-s − 1.40·51-s + 1.82·53-s − 2.23·55-s − 0.264·57-s − 0.781·59-s − 1.27·61-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: 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

\(L(1)\) \(\approx\) \(1.247649593\)
\(L(\frac12)\) \(\approx\) \(1.247649593\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T + 3T^{2} \)
5 \( 1 + 3.31T + 5T^{2} \)
7 \( 1 - 3.31T + 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
23 \( 1 - 6.63T + 23T^{2} \)
29 \( 1 + 6.63T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6.63T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + 9.94T + 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 9.94T + 61T^{2} \)
67 \( 1 + 8T + 67T^{2} \)
71 \( 1 + 6.63T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 4T + 89T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−8.067205164676451621213300957554, −7.56091010996316573565695497531, −6.90242670636169450566330297932, −6.03555115459770356723542639384, −5.25762982876777096231523374680, −4.60611409192651308942943323164, −3.95544101904848369680286539119, −3.12131019385518060962220064642, −1.45119405589658494255691524224, −0.73517404529325591849329543340, 0.73517404529325591849329543340, 1.45119405589658494255691524224, 3.12131019385518060962220064642, 3.95544101904848369680286539119, 4.60611409192651308942943323164, 5.25762982876777096231523374680, 6.03555115459770356723542639384, 6.90242670636169450566330297932, 7.56091010996316573565695497531, 8.067205164676451621213300957554

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