Properties

Degree 1
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.995 - 0.0943i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.422 − 0.906i)5-s + (0.906 + 0.422i)11-s + (0.0871 − 0.996i)13-s + (−0.5 − 0.866i)17-s + (−0.258 − 0.965i)19-s + (−0.984 + 0.173i)23-s + (−0.642 + 0.766i)25-s + (0.996 − 0.0871i)29-s + (0.173 + 0.984i)31-s + (−0.258 + 0.965i)37-s + (0.642 + 0.766i)41-s + (0.906 + 0.422i)43-s + (−0.173 + 0.984i)47-s + (0.707 + 0.707i)53-s i·55-s + ⋯
L(s,χ)  = 1  + (−0.422 − 0.906i)5-s + (0.906 + 0.422i)11-s + (0.0871 − 0.996i)13-s + (−0.5 − 0.866i)17-s + (−0.258 − 0.965i)19-s + (−0.984 + 0.173i)23-s + (−0.642 + 0.766i)25-s + (0.996 − 0.0871i)29-s + (0.173 + 0.984i)31-s + (−0.258 + 0.965i)37-s + (0.642 + 0.766i)41-s + (0.906 + 0.422i)43-s + (−0.173 + 0.984i)47-s + (0.707 + 0.707i)53-s i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.995 - 0.0943i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.995 - 0.0943i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.995 - 0.0943i$
motivic weight  =  \(0\)
character  :  $\chi_{6048} (139, \cdot )$
Sato-Tate  :  $\mu(72)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 6048,\ (0:\ ),\ 0.995 - 0.0943i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.558132478 - 0.07370678914i$
$L(\frac12,\chi)$  $\approx$  $1.558132478 - 0.07370678914i$
$L(\chi,1)$  $\approx$  1.012520771 - 0.1460488081i
$L(1,\chi)$  $\approx$  1.012520771 - 0.1460488081i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.641293126253261053295619654189, −17.15493889693931914037862740108, −16.24693494809898702495823774932, −15.88329519591074986896158283480, −14.95378440254847218446561110737, −14.35618271616071148511207358292, −14.078159029172839817699078321112, −13.18898938517684301791065600352, −12.17684020644046354864254087945, −11.85615620458122807220836046406, −11.0926045777668008376340470940, −10.50312222050250312494098317185, −9.84799011623231454416880128696, −8.9633159119422433392686174528, −8.375318393026035040746788434321, −7.636135207641888772488999741, −6.83587754756440892779825243436, −6.271871697461227074582706385887, −5.80855591629272778830672811071, −4.477629791021221188849594435222, −3.87723554073329014657614870278, −3.528642044625462990615842688501, −2.225959537986564805491126467240, −1.89044706992141418148548020909, −0.534398930182619186625709446213, 0.78027984958205406605938827928, 1.32950778265476584091814928321, 2.50824759508536949218197292016, 3.167131985114192024901541577262, 4.368747415465824750215270851195, 4.45864922196616239403654087354, 5.42131204966487296521211929010, 6.17552398951184987883431022367, 7.01470605695240365739959561081, 7.63297436025306490961005510977, 8.49384812111314642249933632473, 8.90024924535488759700032136652, 9.67403123344720054901709408660, 10.32218623683093224233228590165, 11.26707144623029544450625132986, 11.83205698343388401861025172795, 12.36837941328585946203938094327, 13.085782212660087252258824534089, 13.65816012117324449681395198743, 14.44032697741619059318979948792, 15.2025737752911652885831198035, 15.8675102822978914955898528150, 16.18681945746664400635935563348, 17.13726268723981655959289942415, 17.72876185050003328230056120427

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