Properties

Label 2-8048-1.1-c1-0-51
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  + 2.20·3-s − 0.810·5-s − 4.21·7-s + 1.88·9-s − 4.98·11-s + 0.624·13-s − 1.79·15-s − 2.63·17-s + 4.79·19-s − 9.32·21-s + 3.72·23-s − 4.34·25-s − 2.46·27-s − 7.24·29-s + 6.65·31-s − 11.0·33-s + 3.41·35-s + 8.59·37-s + 1.37·39-s − 2.64·41-s − 0.862·43-s − 1.52·45-s + 0.217·47-s + 10.7·49-s − 5.82·51-s − 5.03·53-s + 4.03·55-s + ⋯
L(s)  = 1  + 1.27·3-s − 0.362·5-s − 1.59·7-s + 0.627·9-s − 1.50·11-s + 0.173·13-s − 0.462·15-s − 0.639·17-s + 1.10·19-s − 2.03·21-s + 0.777·23-s − 0.868·25-s − 0.474·27-s − 1.34·29-s + 1.19·31-s − 1.91·33-s + 0.578·35-s + 1.41·37-s + 0.220·39-s − 0.413·41-s − 0.131·43-s − 0.227·45-s + 0.0316·47-s + 1.54·49-s − 0.815·51-s − 0.691·53-s + 0.544·55-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

\(L(1)\) \(\approx\) \(1.646725643\)
\(L(\frac12)\) \(\approx\) \(1.646725643\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.20T + 3T^{2} \)
5 \( 1 + 0.810T + 5T^{2} \)
7 \( 1 + 4.21T + 7T^{2} \)
11 \( 1 + 4.98T + 11T^{2} \)
13 \( 1 - 0.624T + 13T^{2} \)
17 \( 1 + 2.63T + 17T^{2} \)
19 \( 1 - 4.79T + 19T^{2} \)
23 \( 1 - 3.72T + 23T^{2} \)
29 \( 1 + 7.24T + 29T^{2} \)
31 \( 1 - 6.65T + 31T^{2} \)
37 \( 1 - 8.59T + 37T^{2} \)
41 \( 1 + 2.64T + 41T^{2} \)
43 \( 1 + 0.862T + 43T^{2} \)
47 \( 1 - 0.217T + 47T^{2} \)
53 \( 1 + 5.03T + 53T^{2} \)
59 \( 1 - 0.0696T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 - 8.17T + 71T^{2} \)
73 \( 1 + 0.949T + 73T^{2} \)
79 \( 1 + 6.08T + 79T^{2} \)
83 \( 1 - 1.19T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 2.98T + 97T^{2} \)
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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

−7.80293647852482070697540163842, −7.38910143446799205662316182646, −6.56727036824806661224314993450, −5.79304363885466113364727301237, −5.01252603337917631501945440890, −3.94055989812420094658122170935, −3.35220815820814343297032590150, −2.79592816083262135332944535990, −2.17754766930815812995423748739, −0.55549462108227411406038659240, 0.55549462108227411406038659240, 2.17754766930815812995423748739, 2.79592816083262135332944535990, 3.35220815820814343297032590150, 3.94055989812420094658122170935, 5.01252603337917631501945440890, 5.79304363885466113364727301237, 6.56727036824806661224314993450, 7.38910143446799205662316182646, 7.80293647852482070697540163842

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