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Properties

Label 1-6048-6048.2021-r0-0-0
Degree $1$
Conductor $6048$
Sign $-0.735 + 0.677i$
Analytic cond. $28.0867$
Root an. cond. $28.0867$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.573 + 0.819i)5-s + (0.819 − 0.573i)11-s + (−0.0871 + 0.996i)13-s − 17-s + (−0.707 + 0.707i)19-s + (0.642 + 0.766i)23-s + (−0.342 − 0.939i)25-s + (0.996 − 0.0871i)29-s + (0.939 + 0.342i)31-s + (−0.965 + 0.258i)37-s + (−0.642 − 0.766i)41-s + (0.906 + 0.422i)43-s + (0.939 − 0.342i)47-s + (−0.965 + 0.258i)53-s − i·55-s + ⋯
L(s)  = 1  + (−0.573 + 0.819i)5-s + (0.819 − 0.573i)11-s + (−0.0871 + 0.996i)13-s − 17-s + (−0.707 + 0.707i)19-s + (0.642 + 0.766i)23-s + (−0.342 − 0.939i)25-s + (0.996 − 0.0871i)29-s + (0.939 + 0.342i)31-s + (−0.965 + 0.258i)37-s + (−0.642 − 0.766i)41-s + (0.906 + 0.422i)43-s + (0.939 − 0.342i)47-s + (−0.965 + 0.258i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $-0.735 + 0.677i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (2021, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ -0.735 + 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4050663496 + 1.036830727i\)
\(L(\frac12)\) \(\approx\) \(0.4050663496 + 1.036830727i\)
\(L(1)\) \(\approx\) \(0.8842336757 + 0.2719380555i\)
\(L(1)\) \(\approx\) \(0.8842336757 + 0.2719380555i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.573 + 0.819i)T \)
11 \( 1 + (0.819 - 0.573i)T \)
13 \( 1 + (-0.0871 + 0.996i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (0.996 - 0.0871i)T \)
31 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (-0.642 - 0.766i)T \)
43 \( 1 + (0.906 + 0.422i)T \)
47 \( 1 + (0.939 - 0.342i)T \)
53 \( 1 + (-0.965 + 0.258i)T \)
59 \( 1 + (0.996 + 0.0871i)T \)
61 \( 1 + (-0.422 + 0.906i)T \)
67 \( 1 + (-0.819 - 0.573i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.996 - 0.0871i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.939 - 0.342i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.33930848170617859938993362865, −17.017323463901769017079554337249, −15.97875783391038575847583951771, −15.53611365603394194502666532728, −15.00948090698901206401455648040, −14.23156224757449984308709971513, −13.32346599355568676088333522617, −12.83148413839354964700266983767, −12.23085716534451955386460564677, −11.63333023116519947175116713342, −10.83168793639045731178019377227, −10.23237076152875489671863414462, −9.26533338075658094669224973992, −8.76911278035233844314085449749, −8.22602816521230812659182484849, −7.37060240112430450028880132411, −6.690392371818960119365032533735, −6.020938184379046629213512713108, −4.79493371619841282924000578367, −4.713832870991079100557692642749, −3.83943576248077723415031838036, −2.92843541959627007766825956329, −2.10944515702297430080951318482, −1.10835231780389252187095246205, −0.330490618049481596727617135515, 1.04134895491973629021620600956, 1.98844746648118480944902582944, 2.77371698960153003079072147395, 3.6283646477951315325967586766, 4.14511733489952714163110288679, 4.878938593611350325148617258437, 6.03590005087293088008323552882, 6.572706555162703681597938797091, 7.03667763619644259081453835571, 7.89579190407053910687221909983, 8.733907987931082115010352183499, 9.10097279551274845783108221334, 10.18585872746292631574970128570, 10.68798575021804628414959630759, 11.45384666503324418625817618215, 11.86373765144973526930409022141, 12.529761202599834035537170943477, 13.71243309877738097721133749753, 13.92284284757242368606342981256, 14.68286547513100537196605318924, 15.37584580973657740274722665511, 15.8720351441672326437858747808, 16.66823320522545275114822815008, 17.32026069051049312475659069282, 17.89203970125314866996544650361

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