Properties

Label 2-8048-1.1-c1-0-14
Degree $2$
Conductor $8048$
Sign $1$
Analytic cond. $64.2636$
Root an. cond. $8.01645$
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  − 2.44·3-s − 3.94·5-s − 0.854·7-s + 2.96·9-s + 0.211·11-s + 0.0490·13-s + 9.63·15-s − 1.50·17-s + 2.95·19-s + 2.08·21-s − 3.34·23-s + 10.5·25-s + 0.0925·27-s + 8.69·29-s − 7.84·31-s − 0.517·33-s + 3.37·35-s − 3.52·37-s − 0.119·39-s − 8.41·41-s + 4.95·43-s − 11.6·45-s − 8.48·47-s − 6.26·49-s + 3.66·51-s + 13.8·53-s − 0.836·55-s + ⋯
L(s)  = 1  − 1.40·3-s − 1.76·5-s − 0.322·7-s + 0.987·9-s + 0.0638·11-s + 0.0136·13-s + 2.48·15-s − 0.364·17-s + 0.677·19-s + 0.455·21-s − 0.697·23-s + 2.11·25-s + 0.0178·27-s + 1.61·29-s − 1.40·31-s − 0.0900·33-s + 0.570·35-s − 0.578·37-s − 0.0191·39-s − 1.31·41-s + 0.754·43-s − 1.74·45-s − 1.23·47-s − 0.895·49-s + 0.513·51-s + 1.90·53-s − 0.112·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8048\)    =    \(2^{4} \cdot 503\)
Sign: $1$
Analytic conductor: \(64.2636\)
Root analytic conductor: \(8.01645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8048,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2299889784\)
\(L(\frac12)\) \(\approx\) \(0.2299889784\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 - T \)
good3 \( 1 + 2.44T + 3T^{2} \)
5 \( 1 + 3.94T + 5T^{2} \)
7 \( 1 + 0.854T + 7T^{2} \)
11 \( 1 - 0.211T + 11T^{2} \)
13 \( 1 - 0.0490T + 13T^{2} \)
17 \( 1 + 1.50T + 17T^{2} \)
19 \( 1 - 2.95T + 19T^{2} \)
23 \( 1 + 3.34T + 23T^{2} \)
29 \( 1 - 8.69T + 29T^{2} \)
31 \( 1 + 7.84T + 31T^{2} \)
37 \( 1 + 3.52T + 37T^{2} \)
41 \( 1 + 8.41T + 41T^{2} \)
43 \( 1 - 4.95T + 43T^{2} \)
47 \( 1 + 8.48T + 47T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 1.88T + 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 - 8.33T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 + 8.56T + 89T^{2} \)
97 \( 1 + 2.90T + 97T^{2} \)
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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

−7.71333179385252916177285173513, −6.99521034930409654344595717479, −6.58761969256195098857409778166, −5.71429144533311206049932572679, −4.98022723969785913293685848681, −4.39950014740822631701090523466, −3.66775162012815778092616842006, −2.92300156611236703102559908697, −1.36568958863532111197081402863, −0.27446794495568132097097732937, 0.27446794495568132097097732937, 1.36568958863532111197081402863, 2.92300156611236703102559908697, 3.66775162012815778092616842006, 4.39950014740822631701090523466, 4.98022723969785913293685848681, 5.71429144533311206049932572679, 6.58761969256195098857409778166, 6.99521034930409654344595717479, 7.71333179385252916177285173513

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