Properties

Label 2-4024-1.1-c1-0-82
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 $1$

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

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

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: \(1\)
Selberg data: \((2,\ 4024,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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

−8.016405184049220834543698035651, −7.31748013612045969126080298695, −6.41986473676817464303818539782, −5.90508610614037785175448035967, −5.10224476239511368556794100425, −4.25880209150865698788936732578, −3.91068420504076606395970971317, −2.09324305530817391964174111043, −1.35665293997786108160916676778, 0, 1.35665293997786108160916676778, 2.09324305530817391964174111043, 3.91068420504076606395970971317, 4.25880209150865698788936732578, 5.10224476239511368556794100425, 5.90508610614037785175448035967, 6.41986473676817464303818539782, 7.31748013612045969126080298695, 8.016405184049220834543698035651

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