Properties

Label 2-1008-1.1-c5-0-31
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  − 106.·5-s − 49·7-s − 250.·11-s − 300.·13-s − 2.02e3·17-s + 2.25e3·19-s + 3.09e3·23-s + 8.19e3·25-s + 6.60e3·29-s − 833.·31-s + 5.21e3·35-s + 8.95e3·37-s + 7.20e3·41-s − 1.44e4·43-s − 1.79e4·47-s + 2.40e3·49-s + 1.58e4·53-s + 2.66e4·55-s − 2.67e4·59-s − 2.67e4·61-s + 3.19e4·65-s + 4.44e4·67-s − 2.04e4·71-s + 3.87e4·73-s + 1.22e4·77-s + 6.72e4·79-s − 3.58e4·83-s + ⋯
L(s)  = 1  − 1.90·5-s − 0.377·7-s − 0.623·11-s − 0.493·13-s − 1.69·17-s + 1.43·19-s + 1.21·23-s + 2.62·25-s + 1.45·29-s − 0.155·31-s + 0.719·35-s + 1.07·37-s + 0.669·41-s − 1.19·43-s − 1.18·47-s + 0.142·49-s + 0.773·53-s + 1.18·55-s − 0.998·59-s − 0.922·61-s + 0.938·65-s + 1.21·67-s − 0.480·71-s + 0.850·73-s + 0.235·77-s + 1.21·79-s − 0.571·83-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 \( 1 + 106.T + 3.12e3T^{2} \)
11 \( 1 + 250.T + 1.61e5T^{2} \)
13 \( 1 + 300.T + 3.71e5T^{2} \)
17 \( 1 + 2.02e3T + 1.41e6T^{2} \)
19 \( 1 - 2.25e3T + 2.47e6T^{2} \)
23 \( 1 - 3.09e3T + 6.43e6T^{2} \)
29 \( 1 - 6.60e3T + 2.05e7T^{2} \)
31 \( 1 + 833.T + 2.86e7T^{2} \)
37 \( 1 - 8.95e3T + 6.93e7T^{2} \)
41 \( 1 - 7.20e3T + 1.15e8T^{2} \)
43 \( 1 + 1.44e4T + 1.47e8T^{2} \)
47 \( 1 + 1.79e4T + 2.29e8T^{2} \)
53 \( 1 - 1.58e4T + 4.18e8T^{2} \)
59 \( 1 + 2.67e4T + 7.14e8T^{2} \)
61 \( 1 + 2.67e4T + 8.44e8T^{2} \)
67 \( 1 - 4.44e4T + 1.35e9T^{2} \)
71 \( 1 + 2.04e4T + 1.80e9T^{2} \)
73 \( 1 - 3.87e4T + 2.07e9T^{2} \)
79 \( 1 - 6.72e4T + 3.07e9T^{2} \)
83 \( 1 + 3.58e4T + 3.93e9T^{2} \)
89 \( 1 + 1.06e5T + 5.58e9T^{2} \)
97 \( 1 - 9.81e4T + 8.58e9T^{2} \)
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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

−8.673426650838941802186296450483, −7.933936027181014797261657516363, −7.22114167522106885735100642756, −6.55974272170554742626207304015, −5.00422136167680257886338900566, −4.49046138470818933952966955484, −3.37137892573862412966444770528, −2.68470075078786629601185064226, −0.861559162196326740773649012234, 0, 0.861559162196326740773649012234, 2.68470075078786629601185064226, 3.37137892573862412966444770528, 4.49046138470818933952966955484, 5.00422136167680257886338900566, 6.55974272170554742626207304015, 7.22114167522106885735100642756, 7.933936027181014797261657516363, 8.673426650838941802186296450483

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