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  =  $(2,\ 6048,\ (\ :1/2),\ -0.707 - 0.707i)$
$L(1)$  $\approx$  $1.855537748$
$L(\frac12)$  $\approx$  $1.855537748$
$L(\frac{3}{2})$   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

−8.240008460882572448064002997736, −7.52479345507043753838683886744, −6.87642533919416638771118416611, −6.15458247193007021086733484689, −5.90106116785724778838169023520, −4.55719865555127846000584211421, −3.79424020855216305143831796140, −3.18911033557827748992604948350, −2.26396220984973643585930574238, −1.33369040534124816482429740362, 0.49804155593539625233567121315, 1.33188331343395734967348885463, 2.19645680772829071272224434621, 3.66998526683505522543673399574, 4.20983417996664663916172523825, 4.70379756013599263189624298133, 5.75467838466530760268813510162, 6.30098778907493941676346776864, 7.07899500884723287863445066205, 7.975890667890795164553207317231

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