Properties

Label 2-1008-1.1-c5-0-54
Degree $2$
Conductor $1008$
Sign $-1$
Analytic cond. $161.666$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.42·5-s − 49·7-s − 91.7·11-s + 224.·13-s − 1.04e3·17-s + 1.01e3·19-s + 2.68e3·23-s − 3.12e3·25-s + 7.65e3·29-s − 8.18e3·31-s + 69.8·35-s + 9.10e3·37-s − 1.83e4·41-s + 2.23e4·43-s − 4.85e3·47-s + 2.40e3·49-s − 940.·53-s + 130.·55-s − 1.31e4·59-s + 2.98e3·61-s − 320.·65-s − 3.70e4·67-s + 7.68e4·71-s − 5.02e4·73-s + 4.49e3·77-s + 1.06e4·79-s + 1.38e4·83-s + ⋯
L(s)  = 1  − 0.0254·5-s − 0.377·7-s − 0.228·11-s + 0.368·13-s − 0.877·17-s + 0.643·19-s + 1.06·23-s − 0.999·25-s + 1.69·29-s − 1.53·31-s + 0.00963·35-s + 1.09·37-s − 1.70·41-s + 1.84·43-s − 0.320·47-s + 0.142·49-s − 0.0459·53-s + 0.00582·55-s − 0.493·59-s + 0.102·61-s − 0.00940·65-s − 1.00·67-s + 1.80·71-s − 1.10·73-s + 0.0863·77-s + 0.191·79-s + 0.219·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(161.666\)
Root analytic conductor: \(12.7148\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 \)
7 \( 1 + 49T \)
good5 \( 1 + 1.42T + 3.12e3T^{2} \)
11 \( 1 + 91.7T + 1.61e5T^{2} \)
13 \( 1 - 224.T + 3.71e5T^{2} \)
17 \( 1 + 1.04e3T + 1.41e6T^{2} \)
19 \( 1 - 1.01e3T + 2.47e6T^{2} \)
23 \( 1 - 2.68e3T + 6.43e6T^{2} \)
29 \( 1 - 7.65e3T + 2.05e7T^{2} \)
31 \( 1 + 8.18e3T + 2.86e7T^{2} \)
37 \( 1 - 9.10e3T + 6.93e7T^{2} \)
41 \( 1 + 1.83e4T + 1.15e8T^{2} \)
43 \( 1 - 2.23e4T + 1.47e8T^{2} \)
47 \( 1 + 4.85e3T + 2.29e8T^{2} \)
53 \( 1 + 940.T + 4.18e8T^{2} \)
59 \( 1 + 1.31e4T + 7.14e8T^{2} \)
61 \( 1 - 2.98e3T + 8.44e8T^{2} \)
67 \( 1 + 3.70e4T + 1.35e9T^{2} \)
71 \( 1 - 7.68e4T + 1.80e9T^{2} \)
73 \( 1 + 5.02e4T + 2.07e9T^{2} \)
79 \( 1 - 1.06e4T + 3.07e9T^{2} \)
83 \( 1 - 1.38e4T + 3.93e9T^{2} \)
89 \( 1 + 3.09e4T + 5.58e9T^{2} \)
97 \( 1 - 9.94e4T + 8.58e9T^{2} \)
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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

−8.913265867078084963887365370545, −7.982407382510926884483029870914, −7.10215804941361864651396717081, −6.30458738085416960321553813492, −5.37782880607108450291840597556, −4.40674536640172518765381408593, −3.38618613519397812270572632307, −2.42925035193917241549797484990, −1.17054055087470805048952338698, 0, 1.17054055087470805048952338698, 2.42925035193917241549797484990, 3.38618613519397812270572632307, 4.40674536640172518765381408593, 5.37782880607108450291840597556, 6.30458738085416960321553813492, 7.10215804941361864651396717081, 7.982407382510926884483029870914, 8.913265867078084963887365370545

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