Properties

Label 2-1008-1.1-c5-0-37
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  − 90.4·5-s + 49·7-s − 552.·11-s − 593.·13-s + 1.42e3·17-s − 318.·19-s + 659.·23-s + 5.05e3·25-s + 8.18e3·29-s + 9.59e3·31-s − 4.43e3·35-s + 5.18e3·37-s + 2.19e3·41-s − 7.45e3·43-s − 1.95e4·47-s + 2.40e3·49-s − 3.65e4·53-s + 4.99e4·55-s + 1.63e4·59-s − 1.08e4·61-s + 5.36e4·65-s − 8.03e3·67-s + 5.59e4·71-s − 7.77e4·73-s − 2.70e4·77-s + 3.20e3·79-s + 7.66e4·83-s + ⋯
L(s)  = 1  − 1.61·5-s + 0.377·7-s − 1.37·11-s − 0.973·13-s + 1.19·17-s − 0.202·19-s + 0.260·23-s + 1.61·25-s + 1.80·29-s + 1.79·31-s − 0.611·35-s + 0.622·37-s + 0.203·41-s − 0.615·43-s − 1.29·47-s + 0.142·49-s − 1.78·53-s + 2.22·55-s + 0.611·59-s − 0.374·61-s + 1.57·65-s − 0.218·67-s + 1.31·71-s − 1.70·73-s − 0.520·77-s + 0.0578·79-s + 1.22·83-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 \( 1 + 90.4T + 3.12e3T^{2} \)
11 \( 1 + 552.T + 1.61e5T^{2} \)
13 \( 1 + 593.T + 3.71e5T^{2} \)
17 \( 1 - 1.42e3T + 1.41e6T^{2} \)
19 \( 1 + 318.T + 2.47e6T^{2} \)
23 \( 1 - 659.T + 6.43e6T^{2} \)
29 \( 1 - 8.18e3T + 2.05e7T^{2} \)
31 \( 1 - 9.59e3T + 2.86e7T^{2} \)
37 \( 1 - 5.18e3T + 6.93e7T^{2} \)
41 \( 1 - 2.19e3T + 1.15e8T^{2} \)
43 \( 1 + 7.45e3T + 1.47e8T^{2} \)
47 \( 1 + 1.95e4T + 2.29e8T^{2} \)
53 \( 1 + 3.65e4T + 4.18e8T^{2} \)
59 \( 1 - 1.63e4T + 7.14e8T^{2} \)
61 \( 1 + 1.08e4T + 8.44e8T^{2} \)
67 \( 1 + 8.03e3T + 1.35e9T^{2} \)
71 \( 1 - 5.59e4T + 1.80e9T^{2} \)
73 \( 1 + 7.77e4T + 2.07e9T^{2} \)
79 \( 1 - 3.20e3T + 3.07e9T^{2} \)
83 \( 1 - 7.66e4T + 3.93e9T^{2} \)
89 \( 1 - 8.42e4T + 5.58e9T^{2} \)
97 \( 1 - 1.01e5T + 8.58e9T^{2} \)
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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

−8.404332234211725080814804577153, −7.975210447021038967037422345016, −7.46144238864300555088881737048, −6.37740747398350402240220344410, −4.92979529781255504783900853325, −4.66949229757156607419583284129, −3.32697978225610252534123577396, −2.62416097158728288387396140608, −0.946620708796242120117184656174, 0, 0.946620708796242120117184656174, 2.62416097158728288387396140608, 3.32697978225610252534123577396, 4.66949229757156607419583284129, 4.92979529781255504783900853325, 6.37740747398350402240220344410, 7.46144238864300555088881737048, 7.975210447021038967037422345016, 8.404332234211725080814804577153

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