Properties

Label 2-1008-1.1-c5-0-27
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  − 76·5-s + 49·7-s + 650·11-s + 762·13-s + 556·17-s + 2.45e3·19-s − 2.95e3·23-s + 2.65e3·25-s + 674·29-s + 3.02e3·31-s − 3.72e3·35-s + 7.73e3·37-s + 1.70e4·41-s − 2.18e4·43-s − 2.39e4·47-s + 2.40e3·49-s − 1.55e4·53-s − 4.94e4·55-s + 5.60e3·59-s + 150·61-s − 5.79e4·65-s + 4.37e4·67-s − 3.91e4·71-s − 2.35e4·73-s + 3.18e4·77-s + 1.78e4·79-s + 3.89e4·83-s + ⋯
L(s)  = 1  − 1.35·5-s + 0.377·7-s + 1.61·11-s + 1.25·13-s + 0.466·17-s + 1.55·19-s − 1.16·23-s + 0.848·25-s + 0.148·29-s + 0.565·31-s − 0.513·35-s + 0.928·37-s + 1.58·41-s − 1.80·43-s − 1.58·47-s + 1/7·49-s − 0.762·53-s − 2.20·55-s + 0.209·59-s + 0.00516·61-s − 1.70·65-s + 1.19·67-s − 0.922·71-s − 0.517·73-s + 0.612·77-s + 0.322·79-s + 0.620·83-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

\(L(3)\) \(\approx\) \(2.347688772\)
\(L(\frac12)\) \(\approx\) \(2.347688772\)
\(L(\frac{7}{2})\) 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 \( 1 + 76 T + p^{5} T^{2} \)
11 \( 1 - 650 T + p^{5} T^{2} \)
13 \( 1 - 762 T + p^{5} T^{2} \)
17 \( 1 - 556 T + p^{5} T^{2} \)
19 \( 1 - 2452 T + p^{5} T^{2} \)
23 \( 1 + 2950 T + p^{5} T^{2} \)
29 \( 1 - 674 T + p^{5} T^{2} \)
31 \( 1 - 3024 T + p^{5} T^{2} \)
37 \( 1 - 7730 T + p^{5} T^{2} \)
41 \( 1 - 17016 T + p^{5} T^{2} \)
43 \( 1 + 21836 T + p^{5} T^{2} \)
47 \( 1 + 23940 T + p^{5} T^{2} \)
53 \( 1 + 15594 T + p^{5} T^{2} \)
59 \( 1 - 5608 T + p^{5} T^{2} \)
61 \( 1 - 150 T + p^{5} T^{2} \)
67 \( 1 - 43784 T + p^{5} T^{2} \)
71 \( 1 + 39178 T + p^{5} T^{2} \)
73 \( 1 + 23570 T + p^{5} T^{2} \)
79 \( 1 - 17892 T + p^{5} T^{2} \)
83 \( 1 - 38972 T + p^{5} T^{2} \)
89 \( 1 + 6024 T + p^{5} T^{2} \)
97 \( 1 - 108430 T + p^{5} T^{2} \)
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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

−9.163875421824666596046625869002, −8.204282630914431482119979954399, −7.78299959856145651207462263590, −6.72740674176465840719531575606, −5.93212363662968578263072047748, −4.65510540150638168536651516415, −3.83621852831691997462990360981, −3.28144973492311739458052863252, −1.51138859002014846292361778605, −0.74002396155714142274377867374, 0.74002396155714142274377867374, 1.51138859002014846292361778605, 3.28144973492311739458052863252, 3.83621852831691997462990360981, 4.65510540150638168536651516415, 5.93212363662968578263072047748, 6.72740674176465840719531575606, 7.78299959856145651207462263590, 8.204282630914431482119979954399, 9.163875421824666596046625869002

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