Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.707 + 0.707i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.96i·5-s i·7-s + 4.76·11-s − 1.26·13-s − 0.378i·17-s − 5i·19-s − 1.93·23-s − 3.80·25-s + 4.24i·29-s − 10.4i·31-s + 2.96·35-s + 2.46·37-s − 10.6i·41-s − 4.73i·43-s − 3.20·47-s + ⋯
L(s)  = 1  + 1.32i·5-s − 0.377i·7-s + 1.43·11-s − 0.351·13-s − 0.0919i·17-s − 1.14i·19-s − 0.402·23-s − 0.760·25-s + 0.787i·29-s − 1.87i·31-s + 0.501·35-s + 0.405·37-s − 1.67i·41-s − 0.721i·43-s − 0.467·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.707 + 0.707i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 + 0.707i)$
$L(1)$  $\approx$  $1.747551332$
$L(\frac12)$  $\approx$  $1.747551332$
$L(\frac{3}{2})$   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 \( 1 - 2.96iT - 5T^{2} \)
11 \( 1 - 4.76T + 11T^{2} \)
13 \( 1 + 1.26T + 13T^{2} \)
17 \( 1 + 0.378iT - 17T^{2} \)
19 \( 1 + 5iT - 19T^{2} \)
23 \( 1 + 1.93T + 23T^{2} \)
29 \( 1 - 4.24iT - 29T^{2} \)
31 \( 1 + 10.4iT - 31T^{2} \)
37 \( 1 - 2.46T + 37T^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + 4.73iT - 43T^{2} \)
47 \( 1 + 3.20T + 47T^{2} \)
53 \( 1 + 8.76iT - 53T^{2} \)
59 \( 1 - 3.48T + 59T^{2} \)
61 \( 1 + 4.92T + 61T^{2} \)
67 \( 1 + 7.66iT - 67T^{2} \)
71 \( 1 + 0.240T + 71T^{2} \)
73 \( 1 + 0.196T + 73T^{2} \)
79 \( 1 + 1.80iT - 79T^{2} \)
83 \( 1 - 7.82T + 83T^{2} \)
89 \( 1 - 1.93iT - 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−7.77366506703343517336105968348, −7.11930006615749208937647147579, −6.70246270582561584343158834358, −6.09069365423354511818776610071, −5.11143195950984289809448776190, −4.11293215343910767831589082793, −3.60336209200955570904301723961, −2.68169178109197688005205847913, −1.87367349928948154386784726585, −0.47875312469303823251492868929, 1.12688807482298077541878111338, 1.64136816663485923068009508528, 2.92862919404095821200058885417, 3.94349971315046700356563912683, 4.51554240016273079845437399216, 5.24175679425701473186818272836, 6.05425354985838043138026660756, 6.57816808628782255827449789560, 7.61921361094488690274402441811, 8.316702068094678836154287622550

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