Properties

Label 2-1008-1.1-c5-0-25
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  − 26·5-s + 49·7-s + 664·11-s + 318·13-s − 1.58e3·17-s − 236·19-s + 2.21e3·23-s − 2.44e3·25-s + 4.95e3·29-s + 7.12e3·31-s − 1.27e3·35-s + 4.35e3·37-s − 1.05e4·41-s + 8.45e3·43-s + 5.35e3·47-s + 2.40e3·49-s + 3.33e4·53-s − 1.72e4·55-s − 1.54e4·59-s − 3.67e4·61-s − 8.26e3·65-s − 4.09e4·67-s − 9.09e3·71-s − 7.34e4·73-s + 3.25e4·77-s − 8.94e4·79-s − 6.42e3·83-s + ⋯
L(s)  = 1  − 0.465·5-s + 0.377·7-s + 1.65·11-s + 0.521·13-s − 1.32·17-s − 0.149·19-s + 0.871·23-s − 0.783·25-s + 1.09·29-s + 1.33·31-s − 0.175·35-s + 0.523·37-s − 0.979·41-s + 0.697·43-s + 0.353·47-s + 1/7·49-s + 1.63·53-s − 0.769·55-s − 0.577·59-s − 1.26·61-s − 0.242·65-s − 1.11·67-s − 0.214·71-s − 1.61·73-s + 0.625·77-s − 1.61·79-s − 0.102·83-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

\(L(3)\) \(\approx\) \(2.469748077\)
\(L(\frac12)\) \(\approx\) \(2.469748077\)
\(L(\frac{7}{2})\) 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 \( 1 + 26 T + p^{5} T^{2} \)
11 \( 1 - 664 T + p^{5} T^{2} \)
13 \( 1 - 318 T + p^{5} T^{2} \)
17 \( 1 + 1582 T + p^{5} T^{2} \)
19 \( 1 + 236 T + p^{5} T^{2} \)
23 \( 1 - 2212 T + p^{5} T^{2} \)
29 \( 1 - 4954 T + p^{5} T^{2} \)
31 \( 1 - 7128 T + p^{5} T^{2} \)
37 \( 1 - 4358 T + p^{5} T^{2} \)
41 \( 1 + 10542 T + p^{5} T^{2} \)
43 \( 1 - 8452 T + p^{5} T^{2} \)
47 \( 1 - 5352 T + p^{5} T^{2} \)
53 \( 1 - 33354 T + p^{5} T^{2} \)
59 \( 1 + 15436 T + p^{5} T^{2} \)
61 \( 1 + 36762 T + p^{5} T^{2} \)
67 \( 1 + 40972 T + p^{5} T^{2} \)
71 \( 1 + 9092 T + p^{5} T^{2} \)
73 \( 1 + 73454 T + p^{5} T^{2} \)
79 \( 1 + 89400 T + p^{5} T^{2} \)
83 \( 1 + 6428 T + p^{5} T^{2} \)
89 \( 1 - 122658 T + p^{5} T^{2} \)
97 \( 1 - 21370 T + p^{5} T^{2} \)
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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

−8.899462623234846105889560755907, −8.691582457015526067610405437665, −7.52060661317107124187391710063, −6.65783103280822926164109997247, −6.01841283494525790175373508952, −4.56712410539357099902877223415, −4.13781253438612584796985480471, −2.95814027920086339572692787348, −1.67033753649178701846000737560, −0.71960190169159892503860734565, 0.71960190169159892503860734565, 1.67033753649178701846000737560, 2.95814027920086339572692787348, 4.13781253438612584796985480471, 4.56712410539357099902877223415, 6.01841283494525790175373508952, 6.65783103280822926164109997247, 7.52060661317107124187391710063, 8.691582457015526067610405437665, 8.899462623234846105889560755907

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