Properties

Label 2-1008-1.1-c3-0-36
Degree $2$
Conductor $1008$
Sign $-1$
Analytic cond. $59.4739$
Root an. cond. $7.71193$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.30·5-s − 7·7-s + 48.9·11-s − 2.60·13-s − 136.·17-s − 45.2·19-s − 38.1·23-s − 85.2·25-s − 52.7·29-s + 14.7·31-s − 44.1·35-s + 333.·37-s − 227.·41-s + 398.·43-s − 184.·47-s + 49·49-s − 359.·53-s + 308.·55-s + 99.9·59-s − 674.·61-s − 16.4·65-s + 376.·67-s − 1.18e3·71-s − 735.·73-s − 342.·77-s + 836.·79-s + 293.·83-s + ⋯
L(s)  = 1  + 0.563·5-s − 0.377·7-s + 1.34·11-s − 0.0556·13-s − 1.95·17-s − 0.545·19-s − 0.345·23-s − 0.682·25-s − 0.337·29-s + 0.0856·31-s − 0.213·35-s + 1.48·37-s − 0.865·41-s + 1.41·43-s − 0.572·47-s + 0.142·49-s − 0.932·53-s + 0.755·55-s + 0.220·59-s − 1.41·61-s − 0.0313·65-s + 0.687·67-s − 1.98·71-s − 1.17·73-s − 0.506·77-s + 1.19·79-s + 0.388·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(59.4739\)
Root analytic conductor: \(7.71193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1008,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 7T \)
good5 \( 1 - 6.30T + 125T^{2} \)
11 \( 1 - 48.9T + 1.33e3T^{2} \)
13 \( 1 + 2.60T + 2.19e3T^{2} \)
17 \( 1 + 136.T + 4.91e3T^{2} \)
19 \( 1 + 45.2T + 6.85e3T^{2} \)
23 \( 1 + 38.1T + 1.21e4T^{2} \)
29 \( 1 + 52.7T + 2.43e4T^{2} \)
31 \( 1 - 14.7T + 2.97e4T^{2} \)
37 \( 1 - 333.T + 5.06e4T^{2} \)
41 \( 1 + 227.T + 6.89e4T^{2} \)
43 \( 1 - 398.T + 7.95e4T^{2} \)
47 \( 1 + 184.T + 1.03e5T^{2} \)
53 \( 1 + 359.T + 1.48e5T^{2} \)
59 \( 1 - 99.9T + 2.05e5T^{2} \)
61 \( 1 + 674.T + 2.26e5T^{2} \)
67 \( 1 - 376.T + 3.00e5T^{2} \)
71 \( 1 + 1.18e3T + 3.57e5T^{2} \)
73 \( 1 + 735.T + 3.89e5T^{2} \)
79 \( 1 - 836.T + 4.93e5T^{2} \)
83 \( 1 - 293.T + 5.71e5T^{2} \)
89 \( 1 + 1.29e3T + 7.04e5T^{2} \)
97 \( 1 + 201.T + 9.12e5T^{2} \)
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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

−9.236378989940766534048141790688, −8.557497873834533495646956044863, −7.36882942875971443385206816302, −6.39188239121309759532992275516, −6.06159912957103008621050021437, −4.60665405853406483759528302819, −3.90997272487971659521127227672, −2.53980732672272079458197846216, −1.55937563887268020883397695451, 0, 1.55937563887268020883397695451, 2.53980732672272079458197846216, 3.90997272487971659521127227672, 4.60665405853406483759528302819, 6.06159912957103008621050021437, 6.39188239121309759532992275516, 7.36882942875971443385206816302, 8.557497873834533495646956044863, 9.236378989940766534048141790688

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