Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.587 - 0.809i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.60i·5-s + 7-s + 0.0549i·11-s + 3.75i·13-s + 3.16·17-s − 0.726i·19-s + 7.77·23-s + 2.41·25-s − 2.75i·29-s + 2.14·31-s + 1.60i·35-s + 0.600i·37-s − 4.53·41-s − 6.85i·43-s + 0.744·47-s + ⋯
L(s)  = 1  + 0.718i·5-s + 0.377·7-s + 0.0165i·11-s + 1.04i·13-s + 0.766·17-s − 0.166i·19-s + 1.62·23-s + 0.483·25-s − 0.512i·29-s + 0.385·31-s + 0.271i·35-s + 0.0987i·37-s − 0.708·41-s − 1.04i·43-s + 0.108·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.587 - 0.809i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.587 - 0.809i)$
$L(1)$  $\approx$  $2.265944793$
$L(\frac12)$  $\approx$  $2.265944793$
$L(\frac{3}{2})$   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 \( 1 - T \)
good5 \( 1 - 1.60iT - 5T^{2} \)
11 \( 1 - 0.0549iT - 11T^{2} \)
13 \( 1 - 3.75iT - 13T^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
19 \( 1 + 0.726iT - 19T^{2} \)
23 \( 1 - 7.77T + 23T^{2} \)
29 \( 1 + 2.75iT - 29T^{2} \)
31 \( 1 - 2.14T + 31T^{2} \)
37 \( 1 - 0.600iT - 37T^{2} \)
41 \( 1 + 4.53T + 41T^{2} \)
43 \( 1 + 6.85iT - 43T^{2} \)
47 \( 1 - 0.744T + 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 - 2.58iT - 59T^{2} \)
61 \( 1 + 9.80iT - 61T^{2} \)
67 \( 1 + 12.2iT - 67T^{2} \)
71 \( 1 - 9.19T + 71T^{2} \)
73 \( 1 - 3.85T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 - 13.1iT - 83T^{2} \)
89 \( 1 - 7.90T + 89T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−8.145082882467159366049511964548, −7.33036881674055746346919651476, −6.84974256060496643061432836652, −6.19366917287708576553206493541, −5.22257725293645644857247521574, −4.63549699784130106450219433497, −3.67257188695181009665359375561, −2.93288577578920902563230769855, −2.05302557526945729940375304316, −0.975624234655695822772924480150, 0.73730701825298773828021694141, 1.46351704182576376838518671288, 2.77571377333264352730428717865, 3.41759443972933319320236137574, 4.49238486952342365475248188068, 5.17107979311993452293236270828, 5.57266541334192762760692741992, 6.61313866808229899597215116953, 7.33215788238332706644677169829, 8.078001156970886706635947482455

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