Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $-0.377 + 0.925i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·5-s + (−1 + 2.44i)7-s − 1.41i·11-s − 2.44i·13-s − 5.19·17-s − 7.34i·19-s + 2.82i·23-s − 2.00·25-s − 7.07i·29-s + 2.44i·31-s + (−1.73 + 4.24i)35-s + 5·37-s − 8.66·41-s − 5·43-s − 8.66·47-s + ⋯
L(s)  = 1  + 0.774·5-s + (−0.377 + 0.925i)7-s − 0.426i·11-s − 0.679i·13-s − 1.26·17-s − 1.68i·19-s + 0.589i·23-s − 0.400·25-s − 1.31i·29-s + 0.439i·31-s + (−0.292 + 0.717i)35-s + 0.821·37-s − 1.35·41-s − 0.762·43-s − 1.26·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.377 + 0.925i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1889, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ -0.377 + 0.925i)\)
\(L(1)\)  \(\approx\)  \(1.022352505\)
\(L(\frac12)\)  \(\approx\)  \(1.022352505\)
\(L(\frac{3}{2})\)   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 \( 1 + (1 - 2.44i)T \)
good5 \( 1 - 1.73T + 5T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + 5.19T + 17T^{2} \)
19 \( 1 + 7.34iT - 19T^{2} \)
23 \( 1 - 2.82iT - 23T^{2} \)
29 \( 1 + 7.07iT - 29T^{2} \)
31 \( 1 - 2.44iT - 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + 8.66T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + 8.66T + 47T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 - 8.66T + 59T^{2} \)
61 \( 1 - 2.44iT - 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 13T + 79T^{2} \)
83 \( 1 - 1.73T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 17.1iT - 97T^{2} \)
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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

−8.587849190724769268193008647750, −7.889975080912016958346346304804, −6.63821226351572235266400003419, −6.36716699825144413890056922871, −5.37230710686915281675145381744, −4.86604173409420923556668248055, −3.55708490797448211037227225228, −2.64485509824527501329535462750, −1.95575135540242145309739374218, −0.29673155227346828015419320557, 1.42506369053523215070259667415, 2.22086211762820082792330300084, 3.48785239393292837442239817350, 4.24918047807647133560683929828, 5.04955768075865872459875128840, 6.13996899913554701368300562726, 6.60589848657121452755309574855, 7.34399508069847759334977751317, 8.240405561723184330601229118515, 9.020628125646532001553119287822

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