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  + 3.14i·5-s i·7-s − 1.41·11-s + 0.449·13-s − 2.51i·17-s + 2.44i·19-s + 5.65·23-s − 4.89·25-s − 8.34i·29-s + 1.55i·31-s + 3.14·35-s + 9·37-s − 10.0i·41-s + 8.34i·43-s + 0.460·47-s + ⋯
L(s)  = 1  + 1.40i·5-s − 0.377i·7-s − 0.426·11-s + 0.124·13-s − 0.608i·17-s + 0.561i·19-s + 1.17·23-s − 0.979·25-s − 1.54i·29-s + 0.278i·31-s + 0.531·35-s + 1.47·37-s − 1.57i·41-s + 1.27i·43-s + 0.0672·47-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  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$
$L(1)$  $\approx$  $1.917258748$
$L(\frac12)$  $\approx$  $1.917258748$
$L(\frac{3}{2})$   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 \( 1 - 3.14iT - 5T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 - 0.449T + 13T^{2} \)
17 \( 1 + 2.51iT - 17T^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 + 8.34iT - 29T^{2} \)
31 \( 1 - 1.55iT - 31T^{2} \)
37 \( 1 - 9T + 37T^{2} \)
41 \( 1 + 10.0iT - 41T^{2} \)
43 \( 1 - 8.34iT - 43T^{2} \)
47 \( 1 - 0.460T + 47T^{2} \)
53 \( 1 + 5.02iT - 53T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 + 5.55T + 61T^{2} \)
67 \( 1 - 13.7iT - 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 - 9.79T + 73T^{2} \)
79 \( 1 - 0.348iT - 79T^{2} \)
83 \( 1 - 2.36T + 83T^{2} \)
89 \( 1 - 1.55iT - 89T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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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

−7.924670609996262883176596056451, −7.42545428906150725942398162727, −6.80349092842131115804932901311, −6.16405004093699228066557396953, −5.39931615409438729077859727839, −4.43780418873700357068986016185, −3.63307778956835385829204561790, −2.85217292964628235219593144575, −2.23799074721963651375769684095, −0.793123381242227450985761166101, 0.69583990142221778454823065342, 1.56488273980748129107658878565, 2.64110285411410659860404076209, 3.56562142016074721951258866031, 4.62810736157576331675689202770, 4.98093502809973294038069890211, 5.72323674648306090983604667822, 6.49369588252490970355083506835, 7.40640295827348832231317237007, 8.104819931208426985599871833522

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