Properties

Label 2-4024-1.1-c1-0-66
Degree $2$
Conductor $4024$
Sign $1$
Analytic cond. $32.1318$
Root an. cond. $5.66849$
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  + 1.01·3-s + 3.66·5-s + 1.84·7-s − 1.97·9-s + 2.84·11-s + 4.78·13-s + 3.70·15-s − 7.85·17-s + 1.81·19-s + 1.87·21-s − 5.67·23-s + 8.42·25-s − 5.03·27-s + 7.12·29-s − 2.60·31-s + 2.87·33-s + 6.77·35-s + 4.12·37-s + 4.84·39-s + 10.0·41-s + 4.94·43-s − 7.24·45-s + 10.9·47-s − 3.57·49-s − 7.94·51-s − 12.8·53-s + 10.4·55-s + ⋯
L(s)  = 1  + 0.584·3-s + 1.63·5-s + 0.699·7-s − 0.658·9-s + 0.857·11-s + 1.32·13-s + 0.957·15-s − 1.90·17-s + 0.415·19-s + 0.408·21-s − 1.18·23-s + 1.68·25-s − 0.969·27-s + 1.32·29-s − 0.467·31-s + 0.501·33-s + 1.14·35-s + 0.678·37-s + 0.775·39-s + 1.56·41-s + 0.754·43-s − 1.07·45-s + 1.59·47-s − 0.511·49-s − 1.11·51-s − 1.76·53-s + 1.40·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $1$
Analytic conductor: \(32.1318\)
Root analytic conductor: \(5.66849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4024,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.698934758\)
\(L(\frac12)\) \(\approx\) \(3.698934758\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.01T + 3T^{2} \)
5 \( 1 - 3.66T + 5T^{2} \)
7 \( 1 - 1.84T + 7T^{2} \)
11 \( 1 - 2.84T + 11T^{2} \)
13 \( 1 - 4.78T + 13T^{2} \)
17 \( 1 + 7.85T + 17T^{2} \)
19 \( 1 - 1.81T + 19T^{2} \)
23 \( 1 + 5.67T + 23T^{2} \)
29 \( 1 - 7.12T + 29T^{2} \)
31 \( 1 + 2.60T + 31T^{2} \)
37 \( 1 - 4.12T + 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 - 4.94T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 + 12.8T + 53T^{2} \)
59 \( 1 - 14.0T + 59T^{2} \)
61 \( 1 + 1.38T + 61T^{2} \)
67 \( 1 + 6.14T + 67T^{2} \)
71 \( 1 - 2.57T + 71T^{2} \)
73 \( 1 - 5.14T + 73T^{2} \)
79 \( 1 + 7.49T + 79T^{2} \)
83 \( 1 + 8.68T + 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−8.621736198300086123783541111632, −7.946947712746172013901914512831, −6.77859738717610717107806834661, −6.10073808898428717187134638030, −5.75967013874344342293134725164, −4.61754061065205424972508429569, −3.87715752399051114583790416267, −2.64875664690098555396958460560, −2.07621593190165400611714883016, −1.17374136650908496040003955490, 1.17374136650908496040003955490, 2.07621593190165400611714883016, 2.64875664690098555396958460560, 3.87715752399051114583790416267, 4.61754061065205424972508429569, 5.75967013874344342293134725164, 6.10073808898428717187134638030, 6.77859738717610717107806834661, 7.946947712746172013901914512831, 8.621736198300086123783541111632

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