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Properties

Label 1-3024-3024.1507-r1-0-0
Degree $1$
Conductor $3024$
Sign $0.999 + 0.00934i$
Analytic cond. $324.973$
Root an. cond. $324.973$
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  + (−0.342 + 0.939i)5-s + (−0.342 − 0.939i)11-s + (−0.984 + 0.173i)13-s + 17-s − i·19-s + (0.173 + 0.984i)23-s + (−0.766 − 0.642i)25-s + (0.984 + 0.173i)29-s + (−0.766 + 0.642i)31-s + (0.866 + 0.5i)37-s + (−0.173 − 0.984i)41-s + (−0.642 + 0.766i)43-s + (−0.766 − 0.642i)47-s + (0.866 + 0.5i)53-s + 55-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)5-s + (−0.342 − 0.939i)11-s + (−0.984 + 0.173i)13-s + 17-s − i·19-s + (0.173 + 0.984i)23-s + (−0.766 − 0.642i)25-s + (0.984 + 0.173i)29-s + (−0.766 + 0.642i)31-s + (0.866 + 0.5i)37-s + (−0.173 − 0.984i)41-s + (−0.642 + 0.766i)43-s + (−0.766 − 0.642i)47-s + (0.866 + 0.5i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.00934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $0.999 + 0.00934i$
Analytic conductor: \(324.973\)
Root analytic conductor: \(324.973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (1507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3024,\ (1:\ ),\ 0.999 + 0.00934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.405685783 + 0.006569415649i\)
\(L(\frac12)\) \(\approx\) \(1.405685783 + 0.006569415649i\)
\(L(1)\) \(\approx\) \(0.8980772597 + 0.09510462437i\)
\(L(1)\) \(\approx\) \(0.8980772597 + 0.09510462437i\)

Euler product

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

−18.83475115962729431021359586197, −18.230145642673227083610717132253, −17.36495738105850640062785833935, −16.63650397375855406302135689270, −16.35615566541676089668574028759, −15.25983343165684365443939038005, −14.81181543219944577210433136226, −14.03418186487639499156537454723, −13.01228842719138158669899294725, −12.39131893628324764279264863789, −12.16255840256289459575424492407, −11.14596325981200804152801033519, −10.04069895504715974868159938374, −9.79895145625758690357140115409, −8.83552618166770946573539108923, −7.85897618941794299665268876694, −7.69063277104950509475382063879, −6.56691954557369517001077514596, −5.61300658771136378279158035389, −4.88556085143363818330109196267, −4.356354514902925353790136591318, −3.37479447642401275634917844039, −2.36476028200012485507392748551, −1.50724205318787533418029281420, −0.50993048581196296314272049125, 0.39327502223563168624447497960, 1.53093628345658001046193300294, 2.83778306905055706320524215873, 3.024507737299753536883602855844, 4.06763573130479141630071506716, 5.05793180238371069977207015991, 5.74911760331401541707366915200, 6.69976783440005218908264264145, 7.31169060227419618266019448852, 7.960246420144773314559315998770, 8.82745462003244819427467114711, 9.73324216477813188721448437086, 10.376136226835763644433316254772, 11.10530178022340453819926779498, 11.7281313923895011305591996671, 12.39081536290528160357143490435, 13.45658602425554503764052734209, 13.942413176783206413888875906274, 14.77654687615101911010132161476, 15.23754693477000070660805958279, 16.111217349007232479901067873836, 16.70417593090000924022313937798, 17.59506680346666398441524031731, 18.23673444710672467446007209964, 18.89765158868181744153409352125

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