Properties

Label 2-4864-1.1-c1-0-112
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  + 3.30·3-s + 2.55·5-s + 1.32·7-s + 7.91·9-s + 2.51·11-s − 1.22·13-s + 8.43·15-s + 0.210·17-s + 19-s + 4.37·21-s − 8.11·23-s + 1.51·25-s + 16.2·27-s + 5.97·29-s + 6.01·31-s + 8.31·33-s + 3.38·35-s − 11.2·37-s − 4.05·39-s + 0.996·41-s − 7.83·43-s + 20.2·45-s − 0.910·47-s − 5.24·49-s + 0.696·51-s − 3.32·53-s + 6.42·55-s + ⋯
L(s)  = 1  + 1.90·3-s + 1.14·5-s + 0.500·7-s + 2.63·9-s + 0.759·11-s − 0.340·13-s + 2.17·15-s + 0.0511·17-s + 0.229·19-s + 0.955·21-s − 1.69·23-s + 0.303·25-s + 3.12·27-s + 1.10·29-s + 1.08·31-s + 1.44·33-s + 0.571·35-s − 1.85·37-s − 0.649·39-s + 0.155·41-s − 1.19·43-s + 3.01·45-s − 0.132·47-s − 0.749·49-s + 0.0975·51-s − 0.456·53-s + 0.866·55-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: \(0\)
Selberg data: \((2,\ 4864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.729106000\)
\(L(\frac12)\) \(\approx\) \(5.729106000\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.30T + 3T^{2} \)
5 \( 1 - 2.55T + 5T^{2} \)
7 \( 1 - 1.32T + 7T^{2} \)
11 \( 1 - 2.51T + 11T^{2} \)
13 \( 1 + 1.22T + 13T^{2} \)
17 \( 1 - 0.210T + 17T^{2} \)
23 \( 1 + 8.11T + 23T^{2} \)
29 \( 1 - 5.97T + 29T^{2} \)
31 \( 1 - 6.01T + 31T^{2} \)
37 \( 1 + 11.2T + 37T^{2} \)
41 \( 1 - 0.996T + 41T^{2} \)
43 \( 1 + 7.83T + 43T^{2} \)
47 \( 1 + 0.910T + 47T^{2} \)
53 \( 1 + 3.32T + 53T^{2} \)
59 \( 1 + 2.04T + 59T^{2} \)
61 \( 1 + 10.0T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + 4.50T + 83T^{2} \)
89 \( 1 + 6.64T + 89T^{2} \)
97 \( 1 - 9.92T + 97T^{2} \)
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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

−8.201607430973263964733561706476, −7.925882839488179536046110119263, −6.83033211402327283705668618088, −6.36969394875779352271254788714, −5.18049323785181982774774908735, −4.39815304998872241749584506822, −3.58684431751563495711930436651, −2.76724456580309903407859704780, −1.93214335669004762720158974282, −1.46474033950011484758272994986, 1.46474033950011484758272994986, 1.93214335669004762720158974282, 2.76724456580309903407859704780, 3.58684431751563495711930436651, 4.39815304998872241749584506822, 5.18049323785181982774774908735, 6.36969394875779352271254788714, 6.83033211402327283705668618088, 7.925882839488179536046110119263, 8.201607430973263964733561706476

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