Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.738 + 0.673i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.35·5-s i·7-s − 1.09i·11-s − 4.10i·13-s + 4.74i·17-s + 7.10·19-s − 3.99·23-s + 0.523·25-s − 4.09·29-s + 10.8i·31-s + 2.35i·35-s + 2.27i·37-s + 1.29i·41-s + 8.59·43-s + 6.79·47-s + ⋯
L(s)  = 1  − 1.05·5-s − 0.377i·7-s − 0.328i·11-s − 1.13i·13-s + 1.15i·17-s + 1.62·19-s − 0.832·23-s + 0.104·25-s − 0.761·29-s + 1.95i·31-s + 0.397i·35-s + 0.373i·37-s + 0.201i·41-s + 1.31·43-s + 0.991·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.738 + 0.673i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.738 + 0.673i)$
$L(1)$  $\approx$  $1.322285160$
$L(\frac12)$  $\approx$  $1.322285160$
$L(\frac{3}{2})$   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 \( 1 + iT \)
good5 \( 1 + 2.35T + 5T^{2} \)
11 \( 1 + 1.09iT - 11T^{2} \)
13 \( 1 + 4.10iT - 13T^{2} \)
17 \( 1 - 4.74iT - 17T^{2} \)
19 \( 1 - 7.10T + 19T^{2} \)
23 \( 1 + 3.99T + 23T^{2} \)
29 \( 1 + 4.09T + 29T^{2} \)
31 \( 1 - 10.8iT - 31T^{2} \)
37 \( 1 - 2.27iT - 37T^{2} \)
41 \( 1 - 1.29iT - 41T^{2} \)
43 \( 1 - 8.59T + 43T^{2} \)
47 \( 1 - 6.79T + 47T^{2} \)
53 \( 1 - 0.688T + 53T^{2} \)
59 \( 1 + 7.72iT - 59T^{2} \)
61 \( 1 + 9.70iT - 61T^{2} \)
67 \( 1 + 8.23T + 67T^{2} \)
71 \( 1 - 6.46T + 71T^{2} \)
73 \( 1 - 8.44T + 73T^{2} \)
79 \( 1 + 5.70iT - 79T^{2} \)
83 \( 1 - 10.0iT - 83T^{2} \)
89 \( 1 + 2.54iT - 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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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

−7.949934134446479422696814338029, −7.48720593883005848808280884178, −6.68208722457436300162046843789, −5.74429455379729574437481079400, −5.19729267713379743512240245137, −4.15797003540947517386724929495, −3.55647101179091243918600829094, −2.97376548193653044857026724585, −1.55129339441423012451648881709, −0.51884005113862071979435041433, 0.72231588585870146577850858501, 2.06106146989572748911984590793, 2.87013931214986180606832834728, 4.00187273609041343256605992156, 4.24359240559278725820011933317, 5.35032443847790980205911521870, 5.91785317740669948830660240622, 7.00642126003364570374944410644, 7.51021568420347135688112224746, 7.894538230290211852110445835109

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