Properties

Degree $2$
Conductor $3024$
Sign $0.612 + 0.790i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.63i·5-s + (−1 + 2.44i)7-s + 1.50i·11-s − 5.91i·13-s + 5.14i·17-s + 5.24·19-s + 8.78i·23-s − 8.24·25-s + 8.91·29-s + 3.24·31-s + (8.91 + 3.63i)35-s + 37-s − 6.65i·41-s − 9.37i·43-s + 3.69·47-s + ⋯
L(s)  = 1  − 1.62i·5-s + (−0.377 + 0.925i)7-s + 0.454i·11-s − 1.64i·13-s + 1.24i·17-s + 1.20·19-s + 1.83i·23-s − 1.64·25-s + 1.65·29-s + 0.582·31-s + (1.50 + 0.615i)35-s + 0.164·37-s − 1.03i·41-s − 1.43i·43-s + 0.538·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.612 + 0.790i$
Motivic weight: \(1\)
Character: $\chi_{3024} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ 0.612 + 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.763164659\)
\(L(\frac12)\) \(\approx\) \(1.763164659\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2.44i)T \)
good5 \( 1 + 3.63iT - 5T^{2} \)
11 \( 1 - 1.50iT - 11T^{2} \)
13 \( 1 + 5.91iT - 13T^{2} \)
17 \( 1 - 5.14iT - 17T^{2} \)
19 \( 1 - 5.24T + 19T^{2} \)
23 \( 1 - 8.78iT - 23T^{2} \)
29 \( 1 - 8.91T + 29T^{2} \)
31 \( 1 - 3.24T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + 6.65iT - 41T^{2} \)
43 \( 1 + 9.37iT - 43T^{2} \)
47 \( 1 - 3.69T + 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 + 8.91T + 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 + 10.8iT - 67T^{2} \)
71 \( 1 + 3.63iT - 71T^{2} \)
73 \( 1 + 0.420iT - 73T^{2} \)
79 \( 1 + 4.47iT - 79T^{2} \)
83 \( 1 + 3.69T + 83T^{2} \)
89 \( 1 - 13.9iT - 89T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−8.597275477212669656731876238334, −8.069180502593490868448327875087, −7.30750847042362138117319947931, −5.97917023144389259752198127974, −5.50026934046529369323275649573, −4.98947592767953590934286369230, −3.88180706533867596836353801951, −2.98699389986473856056529370554, −1.73874032363368046500095787202, −0.72943520556827056746071329179, 0.947912533406744085048649154120, 2.61428333783676743648707251509, 2.99540989080402118089023842689, 4.10055976616369200683849288293, 4.76590973573219876831773732063, 6.19862709925334046586336448537, 6.67111178627447986549172547422, 7.10191461028163337905191440791, 7.87936957149566830342269911259, 8.904656575515714657064117304748

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