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  =  $(2,\ 6048,\ (\ :1/2),\ 0.738 + 0.673i)$
$L(1)$  $\approx$  $2.681503959$
$L(\frac12)$  $\approx$  $2.681503959$
$L(\frac{3}{2})$   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

−7.86120616478631759643257401800, −7.25504316961551315649006348333, −6.65762047489725351325122794430, −5.70202921011116522512068100840, −5.19732945371735093676356054116, −4.59457805335263704252716374372, −3.14921503905816629506628907753, −2.93131575714013403095529084168, −1.61222537482193647574363670723, −0.77756276212792733425982193073, 1.08324308001420716943483310063, 1.97263321155274351528443343788, 2.73866407430504875371508400203, 3.73379889381815290459385974289, 4.58040504535866275112770416025, 5.49127551774026968529556371753, 5.97471748036420753789031489989, 6.58635692677520993438860938775, 7.44692779386703947765911859293, 8.180590903899528918395857549704

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