Properties

Label 2-1008-1.1-c5-0-45
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  − 28.7·5-s − 49·7-s − 270.·11-s + 300.·13-s − 613.·17-s + 1.70e3·19-s + 3.18e3·23-s − 2.29e3·25-s − 4.29e3·29-s − 2.02e3·31-s + 1.40e3·35-s + 5.15e3·37-s + 7.14e3·41-s + 1.95e4·43-s + 1.99e4·47-s + 2.40e3·49-s − 3.94e3·53-s + 7.76e3·55-s − 2.97e4·59-s − 5.05e4·61-s − 8.64e3·65-s − 5.05e3·67-s + 3.28e4·71-s − 1.11e4·73-s + 1.32e4·77-s − 8.18e4·79-s + 1.18e5·83-s + ⋯
L(s)  = 1  − 0.514·5-s − 0.377·7-s − 0.673·11-s + 0.493·13-s − 0.514·17-s + 1.08·19-s + 1.25·23-s − 0.735·25-s − 0.949·29-s − 0.379·31-s + 0.194·35-s + 0.618·37-s + 0.663·41-s + 1.61·43-s + 1.32·47-s + 0.142·49-s − 0.193·53-s + 0.346·55-s − 1.11·59-s − 1.73·61-s − 0.253·65-s − 0.137·67-s + 0.773·71-s − 0.244·73-s + 0.254·77-s − 1.47·79-s + 1.88·83-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 \( 1 + 28.7T + 3.12e3T^{2} \)
11 \( 1 + 270.T + 1.61e5T^{2} \)
13 \( 1 - 300.T + 3.71e5T^{2} \)
17 \( 1 + 613.T + 1.41e6T^{2} \)
19 \( 1 - 1.70e3T + 2.47e6T^{2} \)
23 \( 1 - 3.18e3T + 6.43e6T^{2} \)
29 \( 1 + 4.29e3T + 2.05e7T^{2} \)
31 \( 1 + 2.02e3T + 2.86e7T^{2} \)
37 \( 1 - 5.15e3T + 6.93e7T^{2} \)
41 \( 1 - 7.14e3T + 1.15e8T^{2} \)
43 \( 1 - 1.95e4T + 1.47e8T^{2} \)
47 \( 1 - 1.99e4T + 2.29e8T^{2} \)
53 \( 1 + 3.94e3T + 4.18e8T^{2} \)
59 \( 1 + 2.97e4T + 7.14e8T^{2} \)
61 \( 1 + 5.05e4T + 8.44e8T^{2} \)
67 \( 1 + 5.05e3T + 1.35e9T^{2} \)
71 \( 1 - 3.28e4T + 1.80e9T^{2} \)
73 \( 1 + 1.11e4T + 2.07e9T^{2} \)
79 \( 1 + 8.18e4T + 3.07e9T^{2} \)
83 \( 1 - 1.18e5T + 3.93e9T^{2} \)
89 \( 1 - 4.16e4T + 5.58e9T^{2} \)
97 \( 1 - 4.36e4T + 8.58e9T^{2} \)
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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

−9.003377073118348349123027797609, −7.72944461053358297963549532269, −7.40707532720373618194046179732, −6.19934223827698064471699527285, −5.40854320970119463119340355959, −4.34257277677079490080708863135, −3.42034170797955326635739367227, −2.48943303374564776679849230322, −1.08838532640191393328595686667, 0, 1.08838532640191393328595686667, 2.48943303374564776679849230322, 3.42034170797955326635739367227, 4.34257277677079490080708863135, 5.40854320970119463119340355959, 6.19934223827698064471699527285, 7.40707532720373618194046179732, 7.72944461053358297963549532269, 9.003377073118348349123027797609

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