Properties

Degree $2$
Conductor $3024$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.01i·5-s + 2.64·7-s + 2.01i·11-s − 7.34i·17-s + 3.35·19-s − 9.36i·23-s + 0.937·25-s − 2.29·31-s + 5.33i·35-s + 11.9·37-s − 9.36i·41-s + 7.00·49-s − 4.06·55-s + 16.7i·71-s + 5.33i·77-s + ⋯
L(s)  = 1  + 0.901i·5-s + 0.999·7-s + 0.607i·11-s − 1.78i·17-s + 0.769·19-s − 1.95i·23-s + 0.187·25-s − 0.411·31-s + 0.901i·35-s + 1.96·37-s − 1.46i·41-s + 49-s − 0.547·55-s + 1.98i·71-s + 0.607i·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{3024} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.174300633\)
\(L(\frac12)\) \(\approx\) \(2.174300633\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.64T \)
good5 \( 1 - 2.01iT - 5T^{2} \)
11 \( 1 - 2.01iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 7.34iT - 17T^{2} \)
19 \( 1 - 3.35T + 19T^{2} \)
23 \( 1 + 9.36iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 2.29T + 31T^{2} \)
37 \( 1 - 11.9T + 37T^{2} \)
41 \( 1 + 9.36iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 16.7iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 17.4iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.709293562798415746148941870768, −7.83153892407354301956782121206, −7.19124049689124227591731396035, −6.70014165521851156429368740202, −5.57640471948659405973464275448, −4.83482082952123882257171027579, −4.11975739136516895332749291846, −2.81721883695549176045427757706, −2.32283449726634844263267958265, −0.839469142595442730479484440327, 1.09233936792691387122644793982, 1.74909138116278568078884343740, 3.19257421683174542197157749086, 4.09808106434947960461563360132, 4.88292951646813613282425352156, 5.63053136825846376986787888824, 6.22899230271936802238522062419, 7.58419189651088398436629560150, 7.939390635669311453156006399266, 8.678762918622159243868558903760

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