Properties

Label 2-1008-1.1-c5-0-66
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  + 48.4·5-s + 49·7-s − 163.·11-s − 120.·13-s − 78.1·17-s + 2.26e3·19-s − 2.45e3·23-s − 776.·25-s − 6.98e3·29-s − 2.79e3·31-s + 2.37e3·35-s + 9.45e3·37-s − 1.00e4·41-s + 6.93e3·43-s + 1.16e3·47-s + 2.40e3·49-s − 8.56e3·53-s − 7.92e3·55-s + 6.22e3·59-s − 4.19e4·61-s − 5.85e3·65-s − 1.81e3·67-s − 5.68e4·71-s − 4.42e4·73-s − 8.01e3·77-s − 3.49e4·79-s − 3.96e4·83-s + ⋯
L(s)  = 1  + 0.866·5-s + 0.377·7-s − 0.407·11-s − 0.198·13-s − 0.0656·17-s + 1.43·19-s − 0.966·23-s − 0.248·25-s − 1.54·29-s − 0.522·31-s + 0.327·35-s + 1.13·37-s − 0.937·41-s + 0.571·43-s + 0.0769·47-s + 0.142·49-s − 0.418·53-s − 0.353·55-s + 0.232·59-s − 1.44·61-s − 0.171·65-s − 0.0493·67-s − 1.33·71-s − 0.972·73-s − 0.153·77-s − 0.629·79-s − 0.631·83-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 \( 1 - 48.4T + 3.12e3T^{2} \)
11 \( 1 + 163.T + 1.61e5T^{2} \)
13 \( 1 + 120.T + 3.71e5T^{2} \)
17 \( 1 + 78.1T + 1.41e6T^{2} \)
19 \( 1 - 2.26e3T + 2.47e6T^{2} \)
23 \( 1 + 2.45e3T + 6.43e6T^{2} \)
29 \( 1 + 6.98e3T + 2.05e7T^{2} \)
31 \( 1 + 2.79e3T + 2.86e7T^{2} \)
37 \( 1 - 9.45e3T + 6.93e7T^{2} \)
41 \( 1 + 1.00e4T + 1.15e8T^{2} \)
43 \( 1 - 6.93e3T + 1.47e8T^{2} \)
47 \( 1 - 1.16e3T + 2.29e8T^{2} \)
53 \( 1 + 8.56e3T + 4.18e8T^{2} \)
59 \( 1 - 6.22e3T + 7.14e8T^{2} \)
61 \( 1 + 4.19e4T + 8.44e8T^{2} \)
67 \( 1 + 1.81e3T + 1.35e9T^{2} \)
71 \( 1 + 5.68e4T + 1.80e9T^{2} \)
73 \( 1 + 4.42e4T + 2.07e9T^{2} \)
79 \( 1 + 3.49e4T + 3.07e9T^{2} \)
83 \( 1 + 3.96e4T + 3.93e9T^{2} \)
89 \( 1 - 1.26e5T + 5.58e9T^{2} \)
97 \( 1 + 1.45e5T + 8.58e9T^{2} \)
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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

−8.985047645880803731325828853962, −7.83241754764777569878581022713, −7.31090418798391972475280916887, −5.99549121160976253599714928601, −5.53773099293817830516093598625, −4.51904841942047258760632390658, −3.34277304624709977809589753266, −2.22882059911775026323801605395, −1.39776229940196712807930714986, 0, 1.39776229940196712807930714986, 2.22882059911775026323801605395, 3.34277304624709977809589753266, 4.51904841942047258760632390658, 5.53773099293817830516093598625, 5.99549121160976253599714928601, 7.31090418798391972475280916887, 7.83241754764777569878581022713, 8.985047645880803731325828853962

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