Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.258 + 0.965i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.12·5-s i·7-s − 0.397i·11-s − 6.55i·13-s − 2i·17-s + 0.527·19-s + 8.21·23-s + 4.79·25-s + 4.73·29-s − 7.55i·31-s − 3.12i·35-s + 3.35i·37-s + 7.48i·41-s − 1.62·43-s − 10.4·47-s + ⋯
L(s)  = 1  + 1.39·5-s − 0.377i·7-s − 0.119i·11-s − 1.81i·13-s − 0.485i·17-s + 0.121·19-s + 1.71·23-s + 0.959·25-s + 0.878·29-s − 1.35i·31-s − 0.529i·35-s + 0.551i·37-s + 1.16i·41-s − 0.247·43-s − 1.52·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.258 + 0.965i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.258 + 0.965i)$
$L(1)$  $\approx$  $2.665408149$
$L(\frac12)$  $\approx$  $2.665408149$
$L(\frac{3}{2})$   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 \( 1 - 3.12T + 5T^{2} \)
11 \( 1 + 0.397iT - 11T^{2} \)
13 \( 1 + 6.55iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 0.527T + 19T^{2} \)
23 \( 1 - 8.21T + 23T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 + 7.55iT - 31T^{2} \)
37 \( 1 - 3.35iT - 37T^{2} \)
41 \( 1 - 7.48iT - 41T^{2} \)
43 \( 1 + 1.62T + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 + 0.795T + 53T^{2} \)
59 \( 1 + 2.47iT - 59T^{2} \)
61 \( 1 - 11.4iT - 61T^{2} \)
67 \( 1 - 6.55T + 67T^{2} \)
71 \( 1 + 2.21T + 71T^{2} \)
73 \( 1 + 5.75T + 73T^{2} \)
79 \( 1 + 10.2iT - 79T^{2} \)
83 \( 1 - 8.36iT - 83T^{2} \)
89 \( 1 + 5.31iT - 89T^{2} \)
97 \( 1 + 19.1T + 97T^{2} \)
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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

−7.996161403845006617307037565467, −7.14132128386955349209459667060, −6.45268466320436065920339825700, −5.71929738361772347771439174516, −5.20911482429952152931841952431, −4.46547835913755809712308136802, −3.04637826612850192862577202271, −2.84635270026773653283050701686, −1.54508478891026504594513082786, −0.67620212561380455741458993931, 1.32579090498005799832329908452, 1.96520035562596015841153736045, 2.78314527451617909408218542862, 3.79191076287234359227663311870, 4.89517991328264592488040284192, 5.21783720090348525711149121324, 6.30097606688172575276148504550, 6.60893684625020305151068516995, 7.29728645915343565804770231041, 8.532487316979400724536538121637

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