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  + 0.865i·5-s + i·7-s − 5.82·11-s + 0.727·13-s − 7.62i·17-s − 2.29i·19-s − 7.50·23-s + 4.25·25-s − 0.231i·29-s + 10.7i·31-s − 0.865·35-s + 1.64·37-s + 0.229i·41-s − 6.70i·43-s + 11.6·47-s + ⋯
L(s)  = 1  + 0.386i·5-s + 0.377i·7-s − 1.75·11-s + 0.201·13-s − 1.84i·17-s − 0.527i·19-s − 1.56·23-s + 0.850·25-s − 0.0429i·29-s + 1.92i·31-s − 0.146·35-s + 0.270·37-s + 0.0358i·41-s − 1.02i·43-s + 1.70·47-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  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$
$L(1)$  $\approx$  $1.361457566$
$L(\frac12)$  $\approx$  $1.361457566$
$L(\frac{3}{2})$   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 \( 1 - 0.865iT - 5T^{2} \)
11 \( 1 + 5.82T + 11T^{2} \)
13 \( 1 - 0.727T + 13T^{2} \)
17 \( 1 + 7.62iT - 17T^{2} \)
19 \( 1 + 2.29iT - 19T^{2} \)
23 \( 1 + 7.50T + 23T^{2} \)
29 \( 1 + 0.231iT - 29T^{2} \)
31 \( 1 - 10.7iT - 31T^{2} \)
37 \( 1 - 1.64T + 37T^{2} \)
41 \( 1 - 0.229iT - 41T^{2} \)
43 \( 1 + 6.70iT - 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 - 2.80iT - 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 + 9.08T + 61T^{2} \)
67 \( 1 - 11.2iT - 67T^{2} \)
71 \( 1 - 9.76T + 71T^{2} \)
73 \( 1 - 14.5T + 73T^{2} \)
79 \( 1 - 4.39iT - 79T^{2} \)
83 \( 1 - 2.47T + 83T^{2} \)
89 \( 1 - 3.15iT - 89T^{2} \)
97 \( 1 - 1.25T + 97T^{2} \)
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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

−8.188273133586644695169451257159, −7.30695835461346081399233712728, −6.98239787303527391327780320236, −5.93124482702249933316723682595, −5.22190602928173316124309965007, −4.78943489182549532245926713425, −3.59947706413843641130161753224, −2.68149408992611635197967527150, −2.34141231530143083468128664982, −0.72153931383446867246898467328, 0.48790278776181591256111129889, 1.79543203130968586411832015443, 2.54986288804067832488224273367, 3.71735728716634752088495419612, 4.23879371630133519684156962028, 5.17286157579047554972373718887, 5.88290709406343775871611234218, 6.37744884365928746985686000966, 7.57210296932769119652584888277, 8.035246694623170641160627630151

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