Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.727 - 0.686i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.21·5-s i·7-s + 2.42i·11-s + 1.11i·13-s − 0.701i·17-s − 0.938·19-s + 4.30·23-s − 0.102·25-s + 4.26·29-s − 7.02i·31-s + 2.21i·35-s − 3.12i·37-s − 0.157i·41-s − 7.08·43-s − 0.867·47-s + ⋯
L(s)  = 1  − 0.989·5-s − 0.377i·7-s + 0.730i·11-s + 0.309i·13-s − 0.170i·17-s − 0.215·19-s + 0.897·23-s − 0.0204·25-s + 0.791·29-s − 1.26i·31-s + 0.374i·35-s − 0.513i·37-s − 0.0246i·41-s − 1.08·43-s − 0.126·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.727 - 0.686i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.727 - 0.686i)$
$L(1)$  $\approx$  $1.220903645$
$L(\frac12)$  $\approx$  $1.220903645$
$L(\frac{3}{2})$   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 \( 1 + 2.21T + 5T^{2} \)
11 \( 1 - 2.42iT - 11T^{2} \)
13 \( 1 - 1.11iT - 13T^{2} \)
17 \( 1 + 0.701iT - 17T^{2} \)
19 \( 1 + 0.938T + 19T^{2} \)
23 \( 1 - 4.30T + 23T^{2} \)
29 \( 1 - 4.26T + 29T^{2} \)
31 \( 1 + 7.02iT - 31T^{2} \)
37 \( 1 + 3.12iT - 37T^{2} \)
41 \( 1 + 0.157iT - 41T^{2} \)
43 \( 1 + 7.08T + 43T^{2} \)
47 \( 1 + 0.867T + 47T^{2} \)
53 \( 1 - 3.91T + 53T^{2} \)
59 \( 1 - 7.28iT - 59T^{2} \)
61 \( 1 + 4.57iT - 61T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 + 6.93T + 71T^{2} \)
73 \( 1 - 7.02T + 73T^{2} \)
79 \( 1 - 6.65iT - 79T^{2} \)
83 \( 1 - 8.41iT - 83T^{2} \)
89 \( 1 - 7.34iT - 89T^{2} \)
97 \( 1 - 7.99T + 97T^{2} \)
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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

−8.023587435130852096366905495140, −7.45733318370124347887387103404, −6.91973933565898856131698288819, −6.14641540491835142708237354714, −5.11489210390606931807251937937, −4.42295200465695761333512151915, −3.88158006003341174392272502017, −2.97524822454117913474147768559, −1.96913182368824417980503126783, −0.74567762431460259296341890467, 0.45502536183360928145579912174, 1.64102522064385391604740896746, 3.00074963283883144076783486291, 3.34271118428049085010247869062, 4.39419436967559148831533852543, 5.02452416753042013859811436066, 5.88277658121896838080356983303, 6.61461207949011650005726323434, 7.33834794214976718568485363278, 8.075223505880459152386696259892

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