Properties

Label 2-4008-1.1-c1-0-1
Degree $2$
Conductor $4008$
Sign $1$
Analytic cond. $32.0040$
Root an. cond. $5.65721$
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-s − 1.68·5-s − 2.23·7-s + 9-s − 3.03·11-s − 4.55·13-s + 1.68·15-s − 2.98·17-s − 2.14·19-s + 2.23·21-s + 1.24·23-s − 2.16·25-s − 27-s − 6.48·29-s + 0.905·31-s + 3.03·33-s + 3.75·35-s + 5.19·37-s + 4.55·39-s + 0.662·41-s − 10.7·43-s − 1.68·45-s + 6.58·47-s − 2.02·49-s + 2.98·51-s − 8.56·53-s + 5.11·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.753·5-s − 0.843·7-s + 0.333·9-s − 0.914·11-s − 1.26·13-s + 0.435·15-s − 0.724·17-s − 0.493·19-s + 0.486·21-s + 0.259·23-s − 0.432·25-s − 0.192·27-s − 1.20·29-s + 0.162·31-s + 0.528·33-s + 0.635·35-s + 0.853·37-s + 0.729·39-s + 0.103·41-s − 1.63·43-s − 0.251·45-s + 0.960·47-s − 0.289·49-s + 0.418·51-s − 1.17·53-s + 0.689·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $1$
Analytic conductor: \(32.0040\)
Root analytic conductor: \(5.65721\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4008,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2680019752\)
\(L(\frac12)\) \(\approx\) \(0.2680019752\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
167 \( 1 + T \)
good5 \( 1 + 1.68T + 5T^{2} \)
7 \( 1 + 2.23T + 7T^{2} \)
11 \( 1 + 3.03T + 11T^{2} \)
13 \( 1 + 4.55T + 13T^{2} \)
17 \( 1 + 2.98T + 17T^{2} \)
19 \( 1 + 2.14T + 19T^{2} \)
23 \( 1 - 1.24T + 23T^{2} \)
29 \( 1 + 6.48T + 29T^{2} \)
31 \( 1 - 0.905T + 31T^{2} \)
37 \( 1 - 5.19T + 37T^{2} \)
41 \( 1 - 0.662T + 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 - 6.58T + 47T^{2} \)
53 \( 1 + 8.56T + 53T^{2} \)
59 \( 1 + 0.576T + 59T^{2} \)
61 \( 1 + 9.76T + 61T^{2} \)
67 \( 1 - 4.66T + 67T^{2} \)
71 \( 1 - 2.31T + 71T^{2} \)
73 \( 1 - 13.5T + 73T^{2} \)
79 \( 1 - 6.77T + 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + 1.55T + 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−8.222787010360180992982986167118, −7.70441608483835403074660442874, −6.95305054114164193312477176886, −6.33879730173701072322970802565, −5.39850150575523940941213685556, −4.72803287744476540150399478558, −3.92451311055458877585008007118, −2.97880269554946201719042615092, −2.05768441220548396916584711178, −0.28395693868843407814609047578, 0.28395693868843407814609047578, 2.05768441220548396916584711178, 2.97880269554946201719042615092, 3.92451311055458877585008007118, 4.72803287744476540150399478558, 5.39850150575523940941213685556, 6.33879730173701072322970802565, 6.95305054114164193312477176886, 7.70441608483835403074660442874, 8.222787010360180992982986167118

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