Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $-0.968 - 0.248i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.465·5-s + (−0.656 + 2.56i)7-s + 5.28i·11-s + 4.50i·13-s + 3.15·17-s − 1.42i·19-s + 2.27i·23-s − 4.78·25-s − 6.09i·29-s + 2.76i·31-s + (0.305 − 1.19i)35-s − 0.613·37-s − 6.93·41-s − 3.08·43-s + 9.30·47-s + ⋯
L(s)  = 1  − 0.208·5-s + (−0.248 + 0.968i)7-s + 1.59i·11-s + 1.24i·13-s + 0.766·17-s − 0.327i·19-s + 0.474i·23-s − 0.956·25-s − 1.13i·29-s + 0.497i·31-s + (0.0516 − 0.201i)35-s − 0.100·37-s − 1.08·41-s − 0.470·43-s + 1.35·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.968 - 0.248i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1889, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ -0.968 - 0.248i)\)
\(L(1)\)  \(\approx\)  \(0.9405187825\)
\(L(\frac12)\)  \(\approx\)  \(0.9405187825\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (0.656 - 2.56i)T \)
good5 \( 1 + 0.465T + 5T^{2} \)
11 \( 1 - 5.28iT - 11T^{2} \)
13 \( 1 - 4.50iT - 13T^{2} \)
17 \( 1 - 3.15T + 17T^{2} \)
19 \( 1 + 1.42iT - 19T^{2} \)
23 \( 1 - 2.27iT - 23T^{2} \)
29 \( 1 + 6.09iT - 29T^{2} \)
31 \( 1 - 2.76iT - 31T^{2} \)
37 \( 1 + 0.613T + 37T^{2} \)
41 \( 1 + 6.93T + 41T^{2} \)
43 \( 1 + 3.08T + 43T^{2} \)
47 \( 1 - 9.30T + 47T^{2} \)
53 \( 1 + 12.1iT - 53T^{2} \)
59 \( 1 - 3.62T + 59T^{2} \)
61 \( 1 - 2.27iT - 61T^{2} \)
67 \( 1 + 13.3T + 67T^{2} \)
71 \( 1 - 7.38iT - 71T^{2} \)
73 \( 1 + 10.8iT - 73T^{2} \)
79 \( 1 + 1.17T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 - 5.39T + 89T^{2} \)
97 \( 1 - 10.4iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.150800708795795794391766034194, −8.365670001315795373293301015644, −7.47847304850616191128572571153, −6.88993417370929853746762963106, −6.05561224078160979849189041057, −5.19235936843139461943206238683, −4.43189264998195892065874868810, −3.57247200912144244129256134431, −2.37138364825019027054632663739, −1.70202169322486778217128795937, 0.30408497865373321582086250110, 1.27465244600209958627632312032, 2.97392359401181303024623796546, 3.45579647865149687406232936070, 4.32525896951717468155003363136, 5.53319830501503744229042373969, 5.92363557495909475701715877215, 6.96728099623662843573716570879, 7.72865967116827746152631434465, 8.254698228020811576564859632338

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