Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.562 - 0.826i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.09·5-s + i·7-s − 0.961i·11-s + 4.60i·13-s − 3.28i·17-s − 3.66·19-s − 2.40·23-s − 0.618·25-s + 9.31·29-s − 1.84i·31-s + 2.09i·35-s + 2.29i·37-s + 0.314i·41-s + 8.64·43-s − 2.34·47-s + ⋯
L(s)  = 1  + 0.936·5-s + 0.377i·7-s − 0.289i·11-s + 1.27i·13-s − 0.795i·17-s − 0.841·19-s − 0.501·23-s − 0.123·25-s + 1.73·29-s − 0.332i·31-s + 0.353i·35-s + 0.377i·37-s + 0.0491i·41-s + 1.31·43-s − 0.341·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.562 - 0.826i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.562 - 0.826i)$
$L(1)$  $\approx$  $2.200799244$
$L(\frac12)$  $\approx$  $2.200799244$
$L(\frac{3}{2})$   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 \( 1 - 2.09T + 5T^{2} \)
11 \( 1 + 0.961iT - 11T^{2} \)
13 \( 1 - 4.60iT - 13T^{2} \)
17 \( 1 + 3.28iT - 17T^{2} \)
19 \( 1 + 3.66T + 19T^{2} \)
23 \( 1 + 2.40T + 23T^{2} \)
29 \( 1 - 9.31T + 29T^{2} \)
31 \( 1 + 1.84iT - 31T^{2} \)
37 \( 1 - 2.29iT - 37T^{2} \)
41 \( 1 - 0.314iT - 41T^{2} \)
43 \( 1 - 8.64T + 43T^{2} \)
47 \( 1 + 2.34T + 47T^{2} \)
53 \( 1 - 7.73T + 53T^{2} \)
59 \( 1 - 7.99iT - 59T^{2} \)
61 \( 1 - 10.6iT - 61T^{2} \)
67 \( 1 + 4.12T + 67T^{2} \)
71 \( 1 - 6.79T + 71T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 + 1.26iT - 79T^{2} \)
83 \( 1 + 2.99iT - 83T^{2} \)
89 \( 1 - 12.8iT - 89T^{2} \)
97 \( 1 - 4.98T + 97T^{2} \)
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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

−8.311164202483921687771936443211, −7.42105685860072651665249768156, −6.53371015910671190176901722000, −6.20606136766625835636699289604, −5.37300249865623945566725264149, −4.59432881404480745219022767366, −3.86746490521037229307382690061, −2.59684579988527375831143898320, −2.18833129725820534463237532869, −1.03530802913198646808129273590, 0.60825017278081078169402014696, 1.76823993177447260918362245472, 2.53113943323709258571392503020, 3.49766532271911128941500976366, 4.35114196366469723722584722260, 5.16521955066987588302681661868, 5.92204297257466301204403669582, 6.40614368750036677126924379559, 7.21468940240029787969288124455, 8.164932635071784669116394883971

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