Properties

Label 2-1008-1.1-c5-0-72
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  + 99.7·5-s − 49·7-s + 26.8·11-s + 10.9·13-s − 593.·17-s − 1.42e3·19-s − 1.51e3·23-s + 6.82e3·25-s + 2.92e3·29-s − 2.04e3·31-s − 4.88e3·35-s + 2.47e3·37-s − 2.01e4·41-s − 9.58e3·43-s − 2.00e4·47-s + 2.40e3·49-s − 6.55e3·53-s + 2.68e3·55-s − 1.82e4·59-s − 3.51e4·61-s + 1.08e3·65-s + 2.88e4·67-s − 1.71e4·71-s + 2.18e4·73-s − 1.31e3·77-s + 8.76e4·79-s − 7.10e4·83-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.377·7-s + 0.0669·11-s + 0.0178·13-s − 0.498·17-s − 0.906·19-s − 0.597·23-s + 2.18·25-s + 0.645·29-s − 0.382·31-s − 0.674·35-s + 0.297·37-s − 1.87·41-s − 0.790·43-s − 1.32·47-s + 0.142·49-s − 0.320·53-s + 0.119·55-s − 0.681·59-s − 1.20·61-s + 0.0319·65-s + 0.784·67-s − 0.402·71-s + 0.480·73-s − 0.0253·77-s + 1.57·79-s − 1.13·83-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 \( 1 - 99.7T + 3.12e3T^{2} \)
11 \( 1 - 26.8T + 1.61e5T^{2} \)
13 \( 1 - 10.9T + 3.71e5T^{2} \)
17 \( 1 + 593.T + 1.41e6T^{2} \)
19 \( 1 + 1.42e3T + 2.47e6T^{2} \)
23 \( 1 + 1.51e3T + 6.43e6T^{2} \)
29 \( 1 - 2.92e3T + 2.05e7T^{2} \)
31 \( 1 + 2.04e3T + 2.86e7T^{2} \)
37 \( 1 - 2.47e3T + 6.93e7T^{2} \)
41 \( 1 + 2.01e4T + 1.15e8T^{2} \)
43 \( 1 + 9.58e3T + 1.47e8T^{2} \)
47 \( 1 + 2.00e4T + 2.29e8T^{2} \)
53 \( 1 + 6.55e3T + 4.18e8T^{2} \)
59 \( 1 + 1.82e4T + 7.14e8T^{2} \)
61 \( 1 + 3.51e4T + 8.44e8T^{2} \)
67 \( 1 - 2.88e4T + 1.35e9T^{2} \)
71 \( 1 + 1.71e4T + 1.80e9T^{2} \)
73 \( 1 - 2.18e4T + 2.07e9T^{2} \)
79 \( 1 - 8.76e4T + 3.07e9T^{2} \)
83 \( 1 + 7.10e4T + 3.93e9T^{2} \)
89 \( 1 - 3.29e3T + 5.58e9T^{2} \)
97 \( 1 + 1.17e5T + 8.58e9T^{2} \)
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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

−8.968103215725456090416184899856, −8.166018919794018008953378101174, −6.70513701641477196798149418761, −6.39079175457676615414010786262, −5.45912047578932198402559371291, −4.59442957932209133852680486087, −3.23586649282948613285045776665, −2.20211100163689064586046104375, −1.49325078203811499215461357307, 0, 1.49325078203811499215461357307, 2.20211100163689064586046104375, 3.23586649282948613285045776665, 4.59442957932209133852680486087, 5.45912047578932198402559371291, 6.39079175457676615414010786262, 6.70513701641477196798149418761, 8.166018919794018008953378101174, 8.968103215725456090416184899856

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