Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.992 - 0.123i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.26i·5-s − 7-s + 2.02i·11-s + 4.05i·13-s − 1.33·17-s − 3.28i·19-s − 2.25·23-s + 3.38·25-s − 3.58i·29-s + 0.464·31-s − 1.26i·35-s + 11.8i·37-s − 1.73·41-s − 1.70i·43-s + 8.45·47-s + ⋯
L(s)  = 1  + 0.567i·5-s − 0.377·7-s + 0.611i·11-s + 1.12i·13-s − 0.324·17-s − 0.753i·19-s − 0.470·23-s + 0.677·25-s − 0.666i·29-s + 0.0833·31-s − 0.214i·35-s + 1.95i·37-s − 0.271·41-s − 0.259i·43-s + 1.23·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.992 - 0.123i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.992 - 0.123i)$
$L(1)$  $\approx$  $0.7427131901$
$L(\frac12)$  $\approx$  $0.7427131901$
$L(\frac{3}{2})$   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 \( 1 - 1.26iT - 5T^{2} \)
11 \( 1 - 2.02iT - 11T^{2} \)
13 \( 1 - 4.05iT - 13T^{2} \)
17 \( 1 + 1.33T + 17T^{2} \)
19 \( 1 + 3.28iT - 19T^{2} \)
23 \( 1 + 2.25T + 23T^{2} \)
29 \( 1 + 3.58iT - 29T^{2} \)
31 \( 1 - 0.464T + 31T^{2} \)
37 \( 1 - 11.8iT - 37T^{2} \)
41 \( 1 + 1.73T + 41T^{2} \)
43 \( 1 + 1.70iT - 43T^{2} \)
47 \( 1 - 8.45T + 47T^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 + 5.49iT - 59T^{2} \)
61 \( 1 - 6.02iT - 61T^{2} \)
67 \( 1 + 0.381iT - 67T^{2} \)
71 \( 1 + 9.47T + 71T^{2} \)
73 \( 1 + 10.0T + 73T^{2} \)
79 \( 1 + 4.90T + 79T^{2} \)
83 \( 1 - 6.95iT - 83T^{2} \)
89 \( 1 + 17.4T + 89T^{2} \)
97 \( 1 - 7.62T + 97T^{2} \)
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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

−8.512766648342749622648551062908, −7.51093277389506145802708369300, −6.93240382164154027707035349264, −6.47943474640652968467352222271, −5.67186234341485980950325905272, −4.57298648637163183540704570523, −4.21398951265333149853564772402, −3.04776836122034374026288941078, −2.43588291942609430025945072021, −1.37914588328621812047547530707, 0.19652017727126935919229786223, 1.20818997292011351215745811660, 2.39477202526793435012061421698, 3.30665339262533052364293908633, 3.97118154267642639857846369250, 4.94120723183095514718186437815, 5.65651142761909832521500735658, 6.12614902011702610460522968167, 7.13598144064487757296786614078, 7.73823934975913831735190878068

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