Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $-0.654 + 0.755i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + (−2 − 1.73i)7-s − 5.19i·11-s + 3.46i·13-s − 6·17-s − 1.73i·19-s + 5.19i·23-s + 4·25-s − 10.3i·29-s − 5.19i·31-s + (−6 − 5.19i)35-s + 37-s − 3·41-s − 10·43-s + 6·47-s + ⋯
L(s)  = 1  + 1.34·5-s + (−0.755 − 0.654i)7-s − 1.56i·11-s + 0.960i·13-s − 1.45·17-s − 0.397i·19-s + 1.08i·23-s + 0.800·25-s − 1.92i·29-s − 0.933i·31-s + (−1.01 − 0.878i)35-s + 0.164·37-s − 0.468·41-s − 1.52·43-s + 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.654 + 0.755i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1889, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ -0.654 + 0.755i)\)
\(L(1)\)  \(\approx\)  \(1.241958986\)
\(L(\frac12)\)  \(\approx\)  \(1.241958986\)
\(L(\frac{3}{2})\)   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 \( 1 + (2 + 1.73i)T \)
good5 \( 1 - 3T + 5T^{2} \)
11 \( 1 + 5.19iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 - 5.19iT - 23T^{2} \)
29 \( 1 + 10.3iT - 29T^{2} \)
31 \( 1 + 5.19iT - 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 - 5.19iT - 71T^{2} \)
73 \( 1 + 3.46iT - 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−8.657458522502953569830698386143, −7.66888029502150422230762208426, −6.61629398488388570737120462336, −6.28127445935905180683440789933, −5.61578839880169822821976620403, −4.50371149494191360826947695984, −3.65667840947889635022657714151, −2.64552566914310812396839325346, −1.74076340317028418083910345390, −0.34832893972256891121407140849, 1.61341195233487171607866973412, 2.37577536382199931415664425892, 3.16833700260265901994212883162, 4.53735762689570983327516331494, 5.20919025650180824820006485366, 5.98826615237472315433458688777, 6.71640111409935872663977329888, 7.21430312976100682884341875726, 8.610190444332558145458579339150, 8.925613923807683273725080869698

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