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Properties

Label 1-6048-6048.4651-r0-0-0
Degree $1$
Conductor $6048$
Sign $0.907 - 0.419i$
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  + (0.819 − 0.573i)5-s + (0.573 − 0.819i)11-s + (0.996 − 0.0871i)13-s + 17-s + (−0.707 + 0.707i)19-s + (−0.642 + 0.766i)23-s + (0.342 − 0.939i)25-s + (0.0871 − 0.996i)29-s + (−0.939 + 0.342i)31-s + (−0.258 + 0.965i)37-s + (−0.642 + 0.766i)41-s + (0.422 + 0.906i)43-s + (0.939 + 0.342i)47-s + (0.258 − 0.965i)53-s − i·55-s + ⋯
L(s)  = 1  + (0.819 − 0.573i)5-s + (0.573 − 0.819i)11-s + (0.996 − 0.0871i)13-s + 17-s + (−0.707 + 0.707i)19-s + (−0.642 + 0.766i)23-s + (0.342 − 0.939i)25-s + (0.0871 − 0.996i)29-s + (−0.939 + 0.342i)31-s + (−0.258 + 0.965i)37-s + (−0.642 + 0.766i)41-s + (0.422 + 0.906i)43-s + (0.939 + 0.342i)47-s + (0.258 − 0.965i)53-s − i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.419i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
Sign: $0.907 - 0.419i$
Analytic conductor: \(28.0867\)
Root analytic conductor: \(28.0867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6048} (4651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6048,\ (0:\ ),\ 0.907 - 0.419i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.485710739 - 0.5469981181i\)
\(L(\frac12)\) \(\approx\) \(2.485710739 - 0.5469981181i\)
\(L(1)\) \(\approx\) \(1.377981663 - 0.1710763488i\)
\(L(1)\) \(\approx\) \(1.377981663 - 0.1710763488i\)

Euler product

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

−17.80020081904302847576979258614, −17.09299506596460943425820382130, −16.60656404767379000875775520701, −15.722892082689746951353341536884, −15.04613418932574402760567982168, −14.367204446371100861069725342517, −13.99441165582247534315500566102, −13.166043116801323716239876427863, −12.51462575045449698837851122397, −11.90053634193911526869531449534, −10.902239306261317383227126632370, −10.5263640633066844236471476645, −9.872394509504881594515495334564, −8.94646033768203593920760513864, −8.728436945980011576506915708631, −7.37683156394218867131391232240, −7.090108851283006942664827576092, −6.16413717762855563458212630261, −5.73795854152720179499569082171, −4.84611296709186889353483866277, −3.92576959361920346637974412256, −3.36938909642091773029027425061, −2.26066500232509756218579914340, −1.863740981251970919905809922811, −0.833497537477025206696750535212, 0.83303578697377423329067061731, 1.46221003143235390377689241955, 2.20291564128815277922840725414, 3.36865042825373274327389788150, 3.796477125769696976879139649892, 4.7736465572774943311655318745, 5.67073424410546635244091309836, 6.006843417922668810831588585748, 6.64799004976444153927176696034, 7.81087054713907661605207114754, 8.38697479122425842090985693141, 8.90107414405859088631977691576, 9.85751442135682817683871287657, 10.09699507441473618470481318588, 11.18563398137216592459469910659, 11.62358175497660459548207542980, 12.5741095929178681420228645605, 13.01565715398497035537450232472, 13.89991063259292835344248028387, 14.113713528019387809267739427984, 14.9769028641015672731955452603, 15.92473872078961337177915517807, 16.36992193064738947456272230919, 17.03103137182160048994289043752, 17.4895703665242394957345086617

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