Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.0828 - 0.996i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.17·5-s + i·7-s − 2.90i·11-s − 1.05i·13-s + 6.48i·17-s + 3.11·19-s + 0.812·23-s − 3.61·25-s − 4.03·29-s + 0.108i·31-s + 1.17i·35-s + 9.97i·37-s + 9.44i·41-s − 10.3·43-s + 4.03·47-s + ⋯
L(s)  = 1  + 0.526·5-s + 0.377i·7-s − 0.876i·11-s − 0.292i·13-s + 1.57i·17-s + 0.714·19-s + 0.169·23-s − 0.722·25-s − 0.749·29-s + 0.0195i·31-s + 0.199i·35-s + 1.63i·37-s + 1.47i·41-s − 1.57·43-s + 0.588·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0828 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0828 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.0828 - 0.996i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.0828 - 0.996i)$
$L(1)$  $\approx$  $1.559017072$
$L(\frac12)$  $\approx$  $1.559017072$
$L(\frac{3}{2})$   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 \( 1 - 1.17T + 5T^{2} \)
11 \( 1 + 2.90iT - 11T^{2} \)
13 \( 1 + 1.05iT - 13T^{2} \)
17 \( 1 - 6.48iT - 17T^{2} \)
19 \( 1 - 3.11T + 19T^{2} \)
23 \( 1 - 0.812T + 23T^{2} \)
29 \( 1 + 4.03T + 29T^{2} \)
31 \( 1 - 0.108iT - 31T^{2} \)
37 \( 1 - 9.97iT - 37T^{2} \)
41 \( 1 - 9.44iT - 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 - 4.03T + 47T^{2} \)
53 \( 1 + 2.97T + 53T^{2} \)
59 \( 1 + 0.868iT - 59T^{2} \)
61 \( 1 + 5.33iT - 61T^{2} \)
67 \( 1 - 3.97T + 67T^{2} \)
71 \( 1 + 0.844T + 71T^{2} \)
73 \( 1 - 3.59T + 73T^{2} \)
79 \( 1 - 9.48iT - 79T^{2} \)
83 \( 1 - 2.71iT - 83T^{2} \)
89 \( 1 + 2.22iT - 89T^{2} \)
97 \( 1 + 3.93T + 97T^{2} \)
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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

−8.254906224892074369703595623606, −7.75518548019607080265908968862, −6.61562329957144242322608847034, −6.12236796648976462545975784810, −5.51634365413219132374463290692, −4.77835022820534296798679423391, −3.66641087881266336421510158088, −3.11628213676369231682291807573, −2.02456135982511011405874266653, −1.18687152758201395290321768583, 0.39436000208568442140695743500, 1.69578577194564960015886786696, 2.41451422544836830823500424195, 3.44732591179153770361503258422, 4.25782226963919375495757773730, 5.11270502341315484627204943046, 5.60466634370029367791265735793, 6.58482465700245207477057926478, 7.39011187377609089263703003887, 7.49381128562352746193424250204

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