Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.331 + 0.943i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.47·5-s i·7-s − 5.11i·11-s − 0.268i·13-s + 1.96i·17-s − 0.909·19-s + 5.86·23-s + 7.04·25-s + 7.19·29-s + 4.57i·31-s + 3.47i·35-s + 6.98i·37-s − 8.34i·41-s + 9.25·43-s − 6.49·47-s + ⋯
L(s)  = 1  − 1.55·5-s − 0.377i·7-s − 1.54i·11-s − 0.0744i·13-s + 0.476i·17-s − 0.208·19-s + 1.22·23-s + 1.40·25-s + 1.33·29-s + 0.821i·31-s + 0.586i·35-s + 1.14i·37-s − 1.30i·41-s + 1.41·43-s − 0.947·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.331 + 0.943i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.331 + 0.943i)$
$L(1)$  $\approx$  $1.008402336$
$L(\frac12)$  $\approx$  $1.008402336$
$L(\frac{3}{2})$   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 \( 1 + 3.47T + 5T^{2} \)
11 \( 1 + 5.11iT - 11T^{2} \)
13 \( 1 + 0.268iT - 13T^{2} \)
17 \( 1 - 1.96iT - 17T^{2} \)
19 \( 1 + 0.909T + 19T^{2} \)
23 \( 1 - 5.86T + 23T^{2} \)
29 \( 1 - 7.19T + 29T^{2} \)
31 \( 1 - 4.57iT - 31T^{2} \)
37 \( 1 - 6.98iT - 37T^{2} \)
41 \( 1 + 8.34iT - 41T^{2} \)
43 \( 1 - 9.25T + 43T^{2} \)
47 \( 1 + 6.49T + 47T^{2} \)
53 \( 1 + 4.54T + 53T^{2} \)
59 \( 1 + 3.50iT - 59T^{2} \)
61 \( 1 - 1.96iT - 61T^{2} \)
67 \( 1 - 11.0T + 67T^{2} \)
71 \( 1 - 1.20T + 71T^{2} \)
73 \( 1 + 9.01T + 73T^{2} \)
79 \( 1 - 13.5iT - 79T^{2} \)
83 \( 1 - 4.96iT - 83T^{2} \)
89 \( 1 + 18.2iT - 89T^{2} \)
97 \( 1 - 3.39T + 97T^{2} \)
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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

−8.026637366567035133079457142305, −7.13882278962795363674084324393, −6.63500820710843416006026860767, −5.71656833844646951801472752754, −4.82303194834991290623362563069, −4.13502096507620962961624990154, −3.36113252532707183525181791735, −2.89897615604120659001505474993, −1.20233454432618521192955890934, −0.35770081260493992756501408230, 0.932396878224518453557256654119, 2.27714370491771953936463962770, 3.06391233349116904779668325911, 4.03497722377004192757488931377, 4.60022190510367603985650284135, 5.15502688607218090606363595637, 6.38708395872268366117455919525, 6.98607815352918935050853073658, 7.68785660770604019534828098608, 8.028596380368862880980069805980

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