Properties

Label 2-1008-1.1-c5-0-61
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  + 28.4·5-s − 49·7-s + 424.·11-s + 508.·13-s + 539.·17-s − 2.60e3·19-s + 261.·23-s − 2.31e3·25-s − 6.87e3·29-s − 5.68e3·31-s − 1.39e3·35-s + 4.90e3·37-s + 5.72e3·41-s + 1.73e3·43-s − 1.01e4·47-s + 2.40e3·49-s + 3.11e4·53-s + 1.20e4·55-s − 3.88e4·59-s + 1.36e4·61-s + 1.44e4·65-s + 3.07e4·67-s − 4.56e4·71-s + 2.17e4·73-s − 2.07e4·77-s − 3.22e4·79-s − 4.66e4·83-s + ⋯
L(s)  = 1  + 0.508·5-s − 0.377·7-s + 1.05·11-s + 0.834·13-s + 0.453·17-s − 1.65·19-s + 0.102·23-s − 0.741·25-s − 1.51·29-s − 1.06·31-s − 0.192·35-s + 0.589·37-s + 0.531·41-s + 0.143·43-s − 0.670·47-s + 0.142·49-s + 1.52·53-s + 0.537·55-s − 1.45·59-s + 0.469·61-s + 0.424·65-s + 0.836·67-s − 1.07·71-s + 0.477·73-s − 0.399·77-s − 0.582·79-s − 0.743·83-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 \( 1 - 28.4T + 3.12e3T^{2} \)
11 \( 1 - 424.T + 1.61e5T^{2} \)
13 \( 1 - 508.T + 3.71e5T^{2} \)
17 \( 1 - 539.T + 1.41e6T^{2} \)
19 \( 1 + 2.60e3T + 2.47e6T^{2} \)
23 \( 1 - 261.T + 6.43e6T^{2} \)
29 \( 1 + 6.87e3T + 2.05e7T^{2} \)
31 \( 1 + 5.68e3T + 2.86e7T^{2} \)
37 \( 1 - 4.90e3T + 6.93e7T^{2} \)
41 \( 1 - 5.72e3T + 1.15e8T^{2} \)
43 \( 1 - 1.73e3T + 1.47e8T^{2} \)
47 \( 1 + 1.01e4T + 2.29e8T^{2} \)
53 \( 1 - 3.11e4T + 4.18e8T^{2} \)
59 \( 1 + 3.88e4T + 7.14e8T^{2} \)
61 \( 1 - 1.36e4T + 8.44e8T^{2} \)
67 \( 1 - 3.07e4T + 1.35e9T^{2} \)
71 \( 1 + 4.56e4T + 1.80e9T^{2} \)
73 \( 1 - 2.17e4T + 2.07e9T^{2} \)
79 \( 1 + 3.22e4T + 3.07e9T^{2} \)
83 \( 1 + 4.66e4T + 3.93e9T^{2} \)
89 \( 1 - 6.37e4T + 5.58e9T^{2} \)
97 \( 1 - 1.15e5T + 8.58e9T^{2} \)
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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

−9.019304493183041706347359607186, −8.051104124478749091926629119294, −6.99958280704156735019284454360, −6.17849038766310963069514310135, −5.62445925909879558583495406493, −4.20027496028757755077374614581, −3.58887889145828041099398340385, −2.21441376989172694708514872592, −1.33628438909052813276167543332, 0, 1.33628438909052813276167543332, 2.21441376989172694708514872592, 3.58887889145828041099398340385, 4.20027496028757755077374614581, 5.62445925909879558583495406493, 6.17849038766310963069514310135, 6.99958280704156735019284454360, 8.051104124478749091926629119294, 9.019304493183041706347359607186

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