Properties

Label 2-1008-1.1-c5-0-65
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 96·5-s − 49·7-s − 720·11-s + 572·13-s − 1.25e3·17-s + 94·19-s + 96·23-s + 6.09e3·25-s + 4.37e3·29-s + 6.24e3·31-s − 4.70e3·35-s − 1.07e4·37-s − 1.20e4·41-s + 9.16e3·43-s − 2.58e4·47-s + 2.40e3·49-s − 1.01e3·53-s − 6.91e4·55-s + 1.24e3·59-s + 7.59e3·61-s + 5.49e4·65-s − 4.11e4·67-s − 3.76e4·71-s − 1.34e4·73-s + 3.52e4·77-s − 6.24e3·79-s − 2.52e4·83-s + ⋯
L(s)  = 1  + 1.71·5-s − 0.377·7-s − 1.79·11-s + 0.938·13-s − 1.05·17-s + 0.0597·19-s + 0.0378·23-s + 1.94·25-s + 0.965·29-s + 1.16·31-s − 0.649·35-s − 1.29·37-s − 1.11·41-s + 0.755·43-s − 1.70·47-s + 1/7·49-s − 0.0495·53-s − 3.08·55-s + 0.0464·59-s + 0.261·61-s + 1.61·65-s − 1.11·67-s − 0.885·71-s − 0.295·73-s + 0.678·77-s − 0.112·79-s − 0.402·83-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: \(1\)
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 \( 1 + p^{2} T \)
good5 \( 1 - 96 T + p^{5} T^{2} \)
11 \( 1 + 720 T + p^{5} T^{2} \)
13 \( 1 - 44 p T + p^{5} T^{2} \)
17 \( 1 + 1254 T + p^{5} T^{2} \)
19 \( 1 - 94 T + p^{5} T^{2} \)
23 \( 1 - 96 T + p^{5} T^{2} \)
29 \( 1 - 4374 T + p^{5} T^{2} \)
31 \( 1 - 6244 T + p^{5} T^{2} \)
37 \( 1 + 10798 T + p^{5} T^{2} \)
41 \( 1 + 12006 T + p^{5} T^{2} \)
43 \( 1 - 9160 T + p^{5} T^{2} \)
47 \( 1 + 25836 T + p^{5} T^{2} \)
53 \( 1 + 1014 T + p^{5} T^{2} \)
59 \( 1 - 1242 T + p^{5} T^{2} \)
61 \( 1 - 7592 T + p^{5} T^{2} \)
67 \( 1 + 41132 T + p^{5} T^{2} \)
71 \( 1 + 37632 T + p^{5} T^{2} \)
73 \( 1 + 13438 T + p^{5} T^{2} \)
79 \( 1 + 6248 T + p^{5} T^{2} \)
83 \( 1 + 25254 T + p^{5} T^{2} \)
89 \( 1 - 45126 T + p^{5} T^{2} \)
97 \( 1 - 107222 T + p^{5} T^{2} \)
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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

−8.830300181827958420312820179025, −8.194848342018045133090884719955, −6.86718655212939747540735479240, −6.22912996534650741085379067927, −5.43415484588931354688228223640, −4.68927453815794082093441911485, −3.09321245068417926461634497036, −2.38254420687638275301604227673, −1.39523458724038252301449199880, 0, 1.39523458724038252301449199880, 2.38254420687638275301604227673, 3.09321245068417926461634497036, 4.68927453815794082093441911485, 5.43415484588931354688228223640, 6.22912996534650741085379067927, 6.86718655212939747540735479240, 8.194848342018045133090884719955, 8.830300181827958420312820179025

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