Properties

Label 2-1008-1.1-c5-0-12
Degree $2$
Conductor $1008$
Sign $1$
Analytic cond. $161.666$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 49·7-s − 240·11-s − 744·13-s + 1.04e3·17-s + 986·19-s + 184·23-s − 3.10e3·25-s + 734·29-s − 5.14e3·31-s + 196·35-s − 6.05e3·37-s − 7.59e3·41-s − 1.30e4·43-s + 1.46e4·47-s + 2.40e3·49-s + 1.45e4·53-s + 960·55-s − 1.33e4·59-s + 9.67e3·61-s + 2.97e3·65-s + 6.21e4·67-s − 2.11e3·71-s − 2.89e4·73-s + 1.17e4·77-s + 1.01e5·79-s − 2.39e4·83-s + ⋯
L(s)  = 1  − 0.0715·5-s − 0.377·7-s − 0.598·11-s − 1.22·13-s + 0.874·17-s + 0.626·19-s + 0.0725·23-s − 0.994·25-s + 0.162·29-s − 0.960·31-s + 0.0270·35-s − 0.727·37-s − 0.705·41-s − 1.07·43-s + 0.968·47-s + 1/7·49-s + 0.710·53-s + 0.0427·55-s − 0.499·59-s + 0.332·61-s + 0.0873·65-s + 1.69·67-s − 0.0497·71-s − 0.634·73-s + 0.226·77-s + 1.83·79-s − 0.381·83-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.272945753\)
\(L(\frac12)\) \(\approx\) \(1.272945753\)
\(L(\frac{7}{2})\) 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 \( 1 + p^{2} T \)
good5 \( 1 + 4 T + p^{5} T^{2} \)
11 \( 1 + 240 T + p^{5} T^{2} \)
13 \( 1 + 744 T + p^{5} T^{2} \)
17 \( 1 - 1042 T + p^{5} T^{2} \)
19 \( 1 - 986 T + p^{5} T^{2} \)
23 \( 1 - 8 p T + p^{5} T^{2} \)
29 \( 1 - 734 T + p^{5} T^{2} \)
31 \( 1 + 5140 T + p^{5} T^{2} \)
37 \( 1 + 6054 T + p^{5} T^{2} \)
41 \( 1 + 7598 T + p^{5} T^{2} \)
43 \( 1 + 13016 T + p^{5} T^{2} \)
47 \( 1 - 14668 T + p^{5} T^{2} \)
53 \( 1 - 274 p T + p^{5} T^{2} \)
59 \( 1 + 13362 T + p^{5} T^{2} \)
61 \( 1 - 9676 T + p^{5} T^{2} \)
67 \( 1 - 62124 T + p^{5} T^{2} \)
71 \( 1 + 2112 T + p^{5} T^{2} \)
73 \( 1 + 28910 T + p^{5} T^{2} \)
79 \( 1 - 101768 T + p^{5} T^{2} \)
83 \( 1 + 23922 T + p^{5} T^{2} \)
89 \( 1 + 141674 T + p^{5} T^{2} \)
97 \( 1 - 99982 T + p^{5} T^{2} \)
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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

−9.430195138450254516778909937783, −8.301299045107961818983842245982, −7.52604794298251094097238424030, −6.86806169375769718311626593268, −5.62188859209303364182612486054, −5.07999533117320304618112769139, −3.82539134430326942690755579859, −2.92304610211004392069886579036, −1.87434744378749569362283149940, −0.47406651125392840568752119963, 0.47406651125392840568752119963, 1.87434744378749569362283149940, 2.92304610211004392069886579036, 3.82539134430326942690755579859, 5.07999533117320304618112769139, 5.62188859209303364182612486054, 6.86806169375769718311626593268, 7.52604794298251094097238424030, 8.301299045107961818983842245982, 9.430195138450254516778909937783

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