Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.888 + 0.458i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.27·5-s i·7-s − 6.57i·11-s − 4.05i·13-s − 6.09i·17-s + 2.08·19-s + 6.85·23-s − 3.37·25-s + 6.53·29-s − 3.26i·31-s + 1.27i·35-s − 2.95i·37-s + 3.35i·41-s − 10.9·43-s + 7.12·47-s + ⋯
L(s)  = 1  − 0.570·5-s − 0.377i·7-s − 1.98i·11-s − 1.12i·13-s − 1.47i·17-s + 0.477·19-s + 1.42·23-s − 0.674·25-s + 1.21·29-s − 0.587i·31-s + 0.215i·35-s − 0.485i·37-s + 0.523i·41-s − 1.67·43-s + 1.03·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.888 + 0.458i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.888 + 0.458i)$
$L(1)$  $\approx$  $1.421336273$
$L(\frac12)$  $\approx$  $1.421336273$
$L(\frac{3}{2})$   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 \( 1 + 1.27T + 5T^{2} \)
11 \( 1 + 6.57iT - 11T^{2} \)
13 \( 1 + 4.05iT - 13T^{2} \)
17 \( 1 + 6.09iT - 17T^{2} \)
19 \( 1 - 2.08T + 19T^{2} \)
23 \( 1 - 6.85T + 23T^{2} \)
29 \( 1 - 6.53T + 29T^{2} \)
31 \( 1 + 3.26iT - 31T^{2} \)
37 \( 1 + 2.95iT - 37T^{2} \)
41 \( 1 - 3.35iT - 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 - 7.12T + 47T^{2} \)
53 \( 1 + 2.87T + 53T^{2} \)
59 \( 1 - 7.75iT - 59T^{2} \)
61 \( 1 + 12.0iT - 61T^{2} \)
67 \( 1 - 3.01T + 67T^{2} \)
71 \( 1 - 3.48T + 71T^{2} \)
73 \( 1 - 2.76T + 73T^{2} \)
79 \( 1 - 0.849iT - 79T^{2} \)
83 \( 1 - 15.8iT - 83T^{2} \)
89 \( 1 + 4.06iT - 89T^{2} \)
97 \( 1 + 4.90T + 97T^{2} \)
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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

−7.960927074092118967494214354004, −7.10290082147748824918382084518, −6.42004621423627613080285786946, −5.47707655336834624195138372751, −5.06588363946822699002741032600, −3.96692489634411990921589066039, −3.15576938080059727742820187560, −2.80164062031246244868119022231, −0.990808915414971665643945458747, −0.43956092862542266239444849789, 1.43419478822141387370926786388, 2.11411613722860564928786729116, 3.20173810945897665328463085952, 4.14832837623549770695583833948, 4.63726992818476901876558715144, 5.38449782755665959330113029090, 6.51145054819298568469842484584, 6.91825070842223244587053212304, 7.60002275663721084625803341203, 8.399809910227191654204032453099

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