Properties

Label 2-8048-1.1-c1-0-23
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  − 3.26·3-s + 1.15·5-s + 1.19·7-s + 7.65·9-s − 4.40·11-s − 5.45·13-s − 3.77·15-s + 1.75·17-s + 0.615·19-s − 3.89·21-s − 3.56·23-s − 3.66·25-s − 15.1·27-s + 5.04·29-s − 3.23·31-s + 14.3·33-s + 1.37·35-s − 8.64·37-s + 17.7·39-s − 6.82·41-s − 4.42·43-s + 8.84·45-s + 4.24·47-s − 5.57·49-s − 5.72·51-s + 9.55·53-s − 5.08·55-s + ⋯
L(s)  = 1  − 1.88·3-s + 0.516·5-s + 0.450·7-s + 2.55·9-s − 1.32·11-s − 1.51·13-s − 0.973·15-s + 0.425·17-s + 0.141·19-s − 0.849·21-s − 0.743·23-s − 0.732·25-s − 2.92·27-s + 0.936·29-s − 0.581·31-s + 2.50·33-s + 0.232·35-s − 1.42·37-s + 2.84·39-s − 1.06·41-s − 0.674·43-s + 1.31·45-s + 0.619·47-s − 0.796·49-s − 0.802·51-s + 1.31·53-s − 0.685·55-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

\(L(1)\) \(\approx\) \(0.5014760444\)
\(L(\frac12)\) \(\approx\) \(0.5014760444\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.26T + 3T^{2} \)
5 \( 1 - 1.15T + 5T^{2} \)
7 \( 1 - 1.19T + 7T^{2} \)
11 \( 1 + 4.40T + 11T^{2} \)
13 \( 1 + 5.45T + 13T^{2} \)
17 \( 1 - 1.75T + 17T^{2} \)
19 \( 1 - 0.615T + 19T^{2} \)
23 \( 1 + 3.56T + 23T^{2} \)
29 \( 1 - 5.04T + 29T^{2} \)
31 \( 1 + 3.23T + 31T^{2} \)
37 \( 1 + 8.64T + 37T^{2} \)
41 \( 1 + 6.82T + 41T^{2} \)
43 \( 1 + 4.42T + 43T^{2} \)
47 \( 1 - 4.24T + 47T^{2} \)
53 \( 1 - 9.55T + 53T^{2} \)
59 \( 1 - 7.62T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 1.28T + 67T^{2} \)
71 \( 1 + 2.48T + 71T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 - 7.25T + 79T^{2} \)
83 \( 1 - 16.6T + 83T^{2} \)
89 \( 1 - 1.65T + 89T^{2} \)
97 \( 1 + 5.02T + 97T^{2} \)
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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

−7.59967006887671663419192613887, −7.05877861068666267262463166997, −6.32858266117070448223225203910, −5.56505507845625064811191511606, −5.10226983128418500381268018950, −4.81872633626071020469465194036, −3.71199474126153180121136933027, −2.39108260079248442113981277888, −1.65300945015597719212050608597, −0.38112023836640495972036648814, 0.38112023836640495972036648814, 1.65300945015597719212050608597, 2.39108260079248442113981277888, 3.71199474126153180121136933027, 4.81872633626071020469465194036, 5.10226983128418500381268018950, 5.56505507845625064811191511606, 6.32858266117070448223225203910, 7.05877861068666267262463166997, 7.59967006887671663419192613887

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