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Properties

Label 1-4032-4032.115-r0-0-0
Degree $1$
Conductor $4032$
Sign $0.396 + 0.918i$
Analytic cond. $18.7245$
Root an. cond. $18.7245$
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  + (0.793 + 0.608i)5-s + (−0.991 − 0.130i)11-s + (0.793 − 0.608i)13-s + (−0.866 + 0.5i)17-s + (0.793 − 0.608i)19-s + (−0.258 + 0.965i)23-s + (0.258 + 0.965i)25-s + (0.991 − 0.130i)29-s − 31-s + (0.130 − 0.991i)37-s + (0.965 − 0.258i)41-s + (−0.608 + 0.793i)43-s − i·47-s + (−0.608 + 0.793i)53-s + (−0.707 − 0.707i)55-s + ⋯
L(s)  = 1  + (0.793 + 0.608i)5-s + (−0.991 − 0.130i)11-s + (0.793 − 0.608i)13-s + (−0.866 + 0.5i)17-s + (0.793 − 0.608i)19-s + (−0.258 + 0.965i)23-s + (0.258 + 0.965i)25-s + (0.991 − 0.130i)29-s − 31-s + (0.130 − 0.991i)37-s + (0.965 − 0.258i)41-s + (−0.608 + 0.793i)43-s − i·47-s + (−0.608 + 0.793i)53-s + (−0.707 − 0.707i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.396 + 0.918i$
Analytic conductor: \(18.7245\)
Root analytic conductor: \(18.7245\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4032,\ (0:\ ),\ 0.396 + 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.494744719 + 0.9825597027i\)
\(L(\frac12)\) \(\approx\) \(1.494744719 + 0.9825597027i\)
\(L(1)\) \(\approx\) \(1.156407444 + 0.2033035165i\)
\(L(1)\) \(\approx\) \(1.156407444 + 0.2033035165i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.793 + 0.608i)T \)
11 \( 1 + (-0.991 - 0.130i)T \)
13 \( 1 + (0.793 - 0.608i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.793 - 0.608i)T \)
23 \( 1 + (-0.258 + 0.965i)T \)
29 \( 1 + (0.991 - 0.130i)T \)
31 \( 1 - T \)
37 \( 1 + (0.130 - 0.991i)T \)
41 \( 1 + (0.965 - 0.258i)T \)
43 \( 1 + (-0.608 + 0.793i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.608 + 0.793i)T \)
59 \( 1 + (0.923 - 0.382i)T \)
61 \( 1 + (-0.382 + 0.923i)T \)
67 \( 1 + (-0.382 + 0.923i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.258 - 0.965i)T \)
79 \( 1 + iT \)
83 \( 1 + (0.130 + 0.991i)T \)
89 \( 1 + (-0.258 - 0.965i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−18.23366896137125460402684437741, −17.85645899651284098900871704041, −16.90829510890905060021165903639, −16.19551378941000003053873268769, −15.92953858760569517701763954683, −14.92313050508616305357704566597, −14.022508755231344440067501819287, −13.62059792922247647152385458003, −12.94164744254358699327486028265, −12.29342105746937046146239669087, −11.47032916434429970077219524099, −10.677569722894222288985020342649, −10.01589433272057402741081302598, −9.35665326267093125576242693918, −8.570675988633921425968734019, −8.0972804947487234097189839451, −6.985559131218900942126235760, −6.37754007161892555803724020182, −5.53763001790218975191208541230, −4.92905109409427086726651465227, −4.21216745963132152604218258653, −3.15820075853432462035798406687, −2.27492176757075744387710460630, −1.62217221634372884714803330479, −0.54588176756534974059172211874, 0.97372269391186126618192056907, 1.95359493089689419292618417788, 2.750452922404171825650207686925, 3.34863056459500614308810071578, 4.365480420756203943268323706729, 5.390773055519928066698020742479, 5.8104820035187627677320133186, 6.58944898454000531090051454280, 7.43322569099119023278973867892, 8.05100737944724745327328757243, 9.01190411092169043609891464318, 9.58086365267626804277024639741, 10.47992958812150347887718075173, 10.89180615021650710736701580316, 11.50513876046605983966818897136, 12.73729036878001308110557425623, 13.13912295744090747631888795511, 13.79428519965690821974299504316, 14.38878136753376658771143696077, 15.40440545426360506653231374071, 15.69032754803690860289224916165, 16.50237716046281606995055389714, 17.57727545025971883082188187892, 17.95188994566672464851366335537, 18.26043450150013902821471777947

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