Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.707 - 0.707i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.24i·5-s i·7-s − 6.07·11-s + 0.0963·13-s − 3.41i·17-s − 5.89i·19-s + 7.54·23-s − 5.56·25-s + 5.81i·29-s + 1.07i·31-s + 3.24·35-s + 7.82·37-s − 0.614i·41-s − 7.75i·43-s + 0.585·47-s + ⋯
L(s)  = 1  + 1.45i·5-s − 0.377i·7-s − 1.83·11-s + 0.0267·13-s − 0.828i·17-s − 1.35i·19-s + 1.57·23-s − 1.11·25-s + 1.07i·29-s + 0.192i·31-s + 0.549·35-s + 1.28·37-s − 0.0958i·41-s − 1.18i·43-s + 0.0854·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.707 - 0.707i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$
$L(1)$  $\approx$  $1.548022118$
$L(\frac12)$  $\approx$  $1.548022118$
$L(\frac{3}{2})$   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 \( 1 - 3.24iT - 5T^{2} \)
11 \( 1 + 6.07T + 11T^{2} \)
13 \( 1 - 0.0963T + 13T^{2} \)
17 \( 1 + 3.41iT - 17T^{2} \)
19 \( 1 + 5.89iT - 19T^{2} \)
23 \( 1 - 7.54T + 23T^{2} \)
29 \( 1 - 5.81iT - 29T^{2} \)
31 \( 1 - 1.07iT - 31T^{2} \)
37 \( 1 - 7.82T + 37T^{2} \)
41 \( 1 + 0.614iT - 41T^{2} \)
43 \( 1 + 7.75iT - 43T^{2} \)
47 \( 1 - 0.585T + 47T^{2} \)
53 \( 1 - 7.39iT - 53T^{2} \)
59 \( 1 + 9.01T + 59T^{2} \)
61 \( 1 + 8.92T + 61T^{2} \)
67 \( 1 - 10.4iT - 67T^{2} \)
71 \( 1 - 9.19T + 71T^{2} \)
73 \( 1 - 11.8T + 73T^{2} \)
79 \( 1 + 11.0iT - 79T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 - 3.01iT - 89T^{2} \)
97 \( 1 - 18.5T + 97T^{2} \)
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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

−7.87741362493891456817013076962, −7.32239841020660246860016097628, −6.97403847626819071813965293664, −6.16645039240651936337803750793, −5.13841283104284438584973739379, −4.76872023318225199712368887438, −3.42664841136651878656592882771, −2.84064677557649676571379900505, −2.37292424376077574886755864900, −0.70143804855573540336420300465, 0.59000010012690945533322683630, 1.66746313885331905129269781127, 2.58790426217531541825395658355, 3.56568024367050850623630559081, 4.62170401685743578915797683343, 5.02388510262064869326466260908, 5.75985837076499415046225789550, 6.32881846679603098267295586294, 7.71121968376370051235180430674, 7.981312263550060297030100655935

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