Properties

Label 2-1008-1.1-c5-0-46
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 78.0·5-s + 49·7-s + 746.·11-s + 9.75·13-s + 1.75e3·17-s + 603.·19-s + 3.17e3·23-s + 2.96e3·25-s + 3.22e3·29-s − 7.88e3·31-s + 3.82e3·35-s + 1.23e4·37-s + 3.69e3·41-s − 1.69e4·43-s − 658.·47-s + 2.40e3·49-s + 2.78e4·53-s + 5.82e4·55-s − 7.45e3·59-s − 4.37e4·61-s + 761.·65-s + 2.86e4·67-s + 9.24e3·71-s − 2.98e4·73-s + 3.65e4·77-s − 3.29e4·79-s − 4.05e4·83-s + ⋯
L(s)  = 1  + 1.39·5-s + 0.377·7-s + 1.85·11-s + 0.0160·13-s + 1.47·17-s + 0.383·19-s + 1.25·23-s + 0.949·25-s + 0.712·29-s − 1.47·31-s + 0.527·35-s + 1.48·37-s + 0.343·41-s − 1.39·43-s − 0.0434·47-s + 0.142·49-s + 1.36·53-s + 2.59·55-s − 0.278·59-s − 1.50·61-s + 0.0223·65-s + 0.779·67-s + 0.217·71-s − 0.654·73-s + 0.703·77-s − 0.593·79-s − 0.645·83-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: \(0\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.536707208\)
\(L(\frac12)\) \(\approx\) \(4.536707208\)
\(L(\frac{7}{2})\) 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 \( 1 - 78.0T + 3.12e3T^{2} \)
11 \( 1 - 746.T + 1.61e5T^{2} \)
13 \( 1 - 9.75T + 3.71e5T^{2} \)
17 \( 1 - 1.75e3T + 1.41e6T^{2} \)
19 \( 1 - 603.T + 2.47e6T^{2} \)
23 \( 1 - 3.17e3T + 6.43e6T^{2} \)
29 \( 1 - 3.22e3T + 2.05e7T^{2} \)
31 \( 1 + 7.88e3T + 2.86e7T^{2} \)
37 \( 1 - 1.23e4T + 6.93e7T^{2} \)
41 \( 1 - 3.69e3T + 1.15e8T^{2} \)
43 \( 1 + 1.69e4T + 1.47e8T^{2} \)
47 \( 1 + 658.T + 2.29e8T^{2} \)
53 \( 1 - 2.78e4T + 4.18e8T^{2} \)
59 \( 1 + 7.45e3T + 7.14e8T^{2} \)
61 \( 1 + 4.37e4T + 8.44e8T^{2} \)
67 \( 1 - 2.86e4T + 1.35e9T^{2} \)
71 \( 1 - 9.24e3T + 1.80e9T^{2} \)
73 \( 1 + 2.98e4T + 2.07e9T^{2} \)
79 \( 1 + 3.29e4T + 3.07e9T^{2} \)
83 \( 1 + 4.05e4T + 3.93e9T^{2} \)
89 \( 1 - 4.13e4T + 5.58e9T^{2} \)
97 \( 1 + 1.10e5T + 8.58e9T^{2} \)
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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

−9.384992210335621116757054033785, −8.629038325855525632269183536767, −7.44178310519713606810169755747, −6.58968175212780053678313279802, −5.82351673965466592760204874527, −5.08013643787039121830022660627, −3.89593371039786226479630275929, −2.83704793396413418114170076076, −1.54399269226190990087192844989, −1.09279383298476658082285008478, 1.09279383298476658082285008478, 1.54399269226190990087192844989, 2.83704793396413418114170076076, 3.89593371039786226479630275929, 5.08013643787039121830022660627, 5.82351673965466592760204874527, 6.58968175212780053678313279802, 7.44178310519713606810169755747, 8.629038325855525632269183536767, 9.384992210335621116757054033785

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