Properties

Label 1-4032-4032.277-r0-0-0
Degree $1$
Conductor $4032$
Sign $0.990 - 0.139i$
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  + (−0.991 + 0.130i)5-s + (−0.793 + 0.608i)11-s + (−0.991 − 0.130i)13-s + (−0.866 + 0.5i)17-s + (0.991 + 0.130i)19-s + (0.965 + 0.258i)23-s + (0.965 − 0.258i)25-s + (−0.793 − 0.608i)29-s − 31-s + (−0.608 − 0.793i)37-s + (−0.258 − 0.965i)41-s + (0.130 + 0.991i)43-s i·47-s + (−0.130 − 0.991i)53-s + (0.707 − 0.707i)55-s + ⋯
L(s)  = 1  + (−0.991 + 0.130i)5-s + (−0.793 + 0.608i)11-s + (−0.991 − 0.130i)13-s + (−0.866 + 0.5i)17-s + (0.991 + 0.130i)19-s + (0.965 + 0.258i)23-s + (0.965 − 0.258i)25-s + (−0.793 − 0.608i)29-s − 31-s + (−0.608 − 0.793i)37-s + (−0.258 − 0.965i)41-s + (0.130 + 0.991i)43-s i·47-s + (−0.130 − 0.991i)53-s + (0.707 − 0.707i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.990 - 0.139i$
Analytic conductor: \(18.7245\)
Root analytic conductor: \(18.7245\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4032,\ (0:\ ),\ 0.990 - 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7491723914 - 0.05242751971i\)
\(L(\frac12)\) \(\approx\) \(0.7491723914 - 0.05242751971i\)
\(L(1)\) \(\approx\) \(0.7218122063 + 0.03482785935i\)
\(L(1)\) \(\approx\) \(0.7218122063 + 0.03482785935i\)

Euler product

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

−18.61821198738189597240008760900, −17.91299387799207686743977686063, −16.95809939301654356208954301023, −16.41870718182616387220451579801, −15.75813986059458821055972005065, −15.17374007823931849841108467545, −14.52827103133852049606602742250, −13.62012855463902008513455297608, −13.016601991994551624709893794746, −12.25938420933150207470897814833, −11.60689513457655739247420798672, −10.97734959380843442781906120949, −10.37679417092559397870588851473, −9.21478637450989113099299266929, −8.90699293847993594901218448198, −7.83949281210012939951666101143, −7.3952807535311990872935223495, −6.76804154090928176015303317095, −5.59339991868380173147465690260, −4.933513467524922923019771750990, −4.37021142681592051956706157731, −3.17778199495146024208880462863, −2.90886194078611632295251186473, −1.66510726336838914261046399709, −0.50790286414232091273441090727, 0.40169382038550331752675622613, 1.77591823092762108274106196404, 2.57680476540745881034799242122, 3.431340962825407962438344621852, 4.15680999647154080864795444513, 5.01616417390532971606788003764, 5.50816216576155522266788501894, 6.797834383105542499023191746794, 7.332357632610385956650763006195, 7.79835128927990202119188445446, 8.69605732087317646658733697826, 9.44627119920621987833837528134, 10.223040830522974355033080660700, 10.97539442071687134157345489670, 11.52797300432119244079372918278, 12.369688329643286548396979759573, 12.8429476117039949974269492581, 13.59222359317013360092929440287, 14.69128296084036233202197491496, 15.00008028145107675709244358291, 15.71345390216337314894636222961, 16.27179720900033304044117529617, 17.16842159467822427281289248890, 17.76140614351649182453196382485, 18.51303301182943733905034003169

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