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  + 2.66i·5-s + i·7-s − 0.164·11-s + 3.56·13-s + 3.41i·17-s − 3.89i·19-s − 5.29·23-s − 2.09·25-s − 7.84i·29-s + 8.72i·31-s − 2.66·35-s + 7.82·37-s + 1.62i·41-s + 11.2i·43-s + 0.585·47-s + ⋯
L(s)  = 1  + 1.19i·5-s + 0.377i·7-s − 0.0496·11-s + 0.987·13-s + 0.828i·17-s − 0.894i·19-s − 1.10·23-s − 0.419·25-s − 1.45i·29-s + 1.56i·31-s − 0.450·35-s + 1.28·37-s + 0.254i·41-s + 1.71i·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  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 - 0.707i)$
$L(1)$  $\approx$  $1.583450791$
$L(\frac12)$  $\approx$  $1.583450791$
$L(\frac{3}{2})$   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 \( 1 - 2.66iT - 5T^{2} \)
11 \( 1 + 0.164T + 11T^{2} \)
13 \( 1 - 3.56T + 13T^{2} \)
17 \( 1 - 3.41iT - 17T^{2} \)
19 \( 1 + 3.89iT - 19T^{2} \)
23 \( 1 + 5.29T + 23T^{2} \)
29 \( 1 + 7.84iT - 29T^{2} \)
31 \( 1 - 8.72iT - 31T^{2} \)
37 \( 1 - 7.82T + 37T^{2} \)
41 \( 1 - 1.62iT - 41T^{2} \)
43 \( 1 - 11.2iT - 43T^{2} \)
47 \( 1 - 0.585T + 47T^{2} \)
53 \( 1 - 14.2iT - 53T^{2} \)
59 \( 1 + 6.98T + 59T^{2} \)
61 \( 1 - 4.92T + 61T^{2} \)
67 \( 1 - 6.83iT - 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 - 5.48T + 73T^{2} \)
79 \( 1 + 9.09iT - 79T^{2} \)
83 \( 1 + 1.95T + 83T^{2} \)
89 \( 1 + 6.05iT - 89T^{2} \)
97 \( 1 - 4.72T + 97T^{2} \)
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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

−8.161856952493609494258654035367, −7.75354463097031516357968559692, −6.73281517409710517799392586214, −6.27285826258043143981171134525, −5.76761488961815591007168285548, −4.60756035188174838147868301073, −3.88666912772314143536615065701, −3.00087245902669297926527335831, −2.41009083856948803308834011849, −1.25279091536722655624917797072, 0.42933589368998279855420924295, 1.33685097044150679318386684295, 2.28316888101350791962345837764, 3.62212109421637180606533628657, 4.04590181763966271908296242441, 5.00390960398924798998189733355, 5.55419762739085094878910836781, 6.32930073923248399633413237953, 7.14316149385782528710324611892, 8.105315570513899256424551781891

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