Properties

Label 2-8048-1.1-c1-0-93
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  − 1.61·3-s − 0.419·5-s + 5.20·7-s − 0.405·9-s − 5.36·11-s + 6.78·13-s + 0.676·15-s + 2.80·17-s + 6.53·19-s − 8.38·21-s + 4.71·23-s − 4.82·25-s + 5.48·27-s + 8.91·29-s − 2.32·31-s + 8.63·33-s − 2.18·35-s + 10.2·37-s − 10.9·39-s − 7.34·41-s + 1.47·43-s + 0.169·45-s − 2.41·47-s + 20.1·49-s − 4.51·51-s − 2.63·53-s + 2.25·55-s + ⋯
L(s)  = 1  − 0.930·3-s − 0.187·5-s + 1.96·7-s − 0.135·9-s − 1.61·11-s + 1.88·13-s + 0.174·15-s + 0.679·17-s + 1.49·19-s − 1.83·21-s + 0.982·23-s − 0.964·25-s + 1.05·27-s + 1.65·29-s − 0.417·31-s + 1.50·33-s − 0.369·35-s + 1.68·37-s − 1.75·39-s − 1.14·41-s + 0.224·43-s + 0.0253·45-s − 0.352·47-s + 2.87·49-s − 0.631·51-s − 0.361·53-s + 0.303·55-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

\(L(1)\) \(\approx\) \(2.005822448\)
\(L(\frac12)\) \(\approx\) \(2.005822448\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.61T + 3T^{2} \)
5 \( 1 + 0.419T + 5T^{2} \)
7 \( 1 - 5.20T + 7T^{2} \)
11 \( 1 + 5.36T + 11T^{2} \)
13 \( 1 - 6.78T + 13T^{2} \)
17 \( 1 - 2.80T + 17T^{2} \)
19 \( 1 - 6.53T + 19T^{2} \)
23 \( 1 - 4.71T + 23T^{2} \)
29 \( 1 - 8.91T + 29T^{2} \)
31 \( 1 + 2.32T + 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 + 7.34T + 41T^{2} \)
43 \( 1 - 1.47T + 43T^{2} \)
47 \( 1 + 2.41T + 47T^{2} \)
53 \( 1 + 2.63T + 53T^{2} \)
59 \( 1 + 9.21T + 59T^{2} \)
61 \( 1 + 0.825T + 61T^{2} \)
67 \( 1 + 5.59T + 67T^{2} \)
71 \( 1 + 4.69T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + 14.6T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 - 9.21T + 89T^{2} \)
97 \( 1 + 6.86T + 97T^{2} \)
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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

−7.904169703315243279593324702496, −7.34221458570456968950551101167, −6.14721309614401189718203793614, −5.70164861488776168746406413999, −4.97699489409679449487250838463, −4.73041777061881722946128621128, −3.50620725418908502472059957445, −2.69034538578258957709791395021, −1.43578117433644985203194367687, −0.834807676463820897916219102771, 0.834807676463820897916219102771, 1.43578117433644985203194367687, 2.69034538578258957709791395021, 3.50620725418908502472059957445, 4.73041777061881722946128621128, 4.97699489409679449487250838463, 5.70164861488776168746406413999, 6.14721309614401189718203793614, 7.34221458570456968950551101167, 7.904169703315243279593324702496

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