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Properties

Label 1-72e2-5184.43-r1-0-0
Degree $1$
Conductor $5184$
Sign $-0.779 + 0.626i$
Analytic cond. $557.098$
Root an. cond. $557.098$
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  + (0.409 + 0.912i)5-s + (0.620 + 0.784i)7-s + (0.631 − 0.775i)11-s + (−0.944 + 0.328i)13-s + (0.342 + 0.939i)17-s + (−0.0436 − 0.999i)19-s + (−0.620 + 0.784i)23-s + (−0.664 + 0.747i)25-s + (0.997 + 0.0726i)29-s + (0.597 + 0.802i)31-s + (−0.461 + 0.887i)35-s + (−0.887 + 0.461i)37-s + (0.979 − 0.202i)41-s + (0.987 + 0.159i)43-s + (0.802 + 0.597i)47-s + ⋯
L(s)  = 1  + (0.409 + 0.912i)5-s + (0.620 + 0.784i)7-s + (0.631 − 0.775i)11-s + (−0.944 + 0.328i)13-s + (0.342 + 0.939i)17-s + (−0.0436 − 0.999i)19-s + (−0.620 + 0.784i)23-s + (−0.664 + 0.747i)25-s + (0.997 + 0.0726i)29-s + (0.597 + 0.802i)31-s + (−0.461 + 0.887i)35-s + (−0.887 + 0.461i)37-s + (0.979 − 0.202i)41-s + (0.987 + 0.159i)43-s + (0.802 + 0.597i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.779 + 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $-0.779 + 0.626i$
Analytic conductor: \(557.098\)
Root analytic conductor: \(557.098\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5184,\ (1:\ ),\ -0.779 + 0.626i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8776452164 + 2.490859539i\)
\(L(\frac12)\) \(\approx\) \(0.8776452164 + 2.490859539i\)
\(L(1)\) \(\approx\) \(1.168704692 + 0.4702036011i\)
\(L(1)\) \(\approx\) \(1.168704692 + 0.4702036011i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.409 + 0.912i)T \)
7 \( 1 + (0.620 + 0.784i)T \)
11 \( 1 + (0.631 - 0.775i)T \)
13 \( 1 + (-0.944 + 0.328i)T \)
17 \( 1 + (0.342 + 0.939i)T \)
19 \( 1 + (-0.0436 - 0.999i)T \)
23 \( 1 + (-0.620 + 0.784i)T \)
29 \( 1 + (0.997 + 0.0726i)T \)
31 \( 1 + (0.597 + 0.802i)T \)
37 \( 1 + (-0.887 + 0.461i)T \)
41 \( 1 + (0.979 - 0.202i)T \)
43 \( 1 + (0.987 + 0.159i)T \)
47 \( 1 + (0.802 + 0.597i)T \)
53 \( 1 + (-0.793 + 0.608i)T \)
59 \( 1 + (0.994 + 0.101i)T \)
61 \( 1 + (-0.717 - 0.696i)T \)
67 \( 1 + (0.653 - 0.756i)T \)
71 \( 1 + (-0.819 + 0.573i)T \)
73 \( 1 + (0.819 + 0.573i)T \)
79 \( 1 + (0.549 + 0.835i)T \)
83 \( 1 + (-0.827 - 0.561i)T \)
89 \( 1 + (0.573 - 0.819i)T \)
97 \( 1 + (0.686 + 0.727i)T \)
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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

−17.62053944196968951007325215508, −16.91144845338294496901871455381, −16.40942234913047456283911342508, −15.69056013963062150155201422430, −14.657663471107157237331076375472, −14.262590019687127413239105046689, −13.69725222773845606840092680204, −12.79296013845688088770989255865, −12.11054965480318067326939219712, −11.87338295813960851408971062468, −10.66979374298389557569495139428, −10.033256094063476364671426742664, −9.597326809460274066861948657454, −8.75069986843371230643461146246, −7.94119722366649503459515764646, −7.47051972408729485483268626634, −6.62544328685472310789570563004, −5.748740576493292829973134211462, −4.97798305644719493135218851540, −4.43179552804477738564278071199, −3.8565502613607807465380083669, −2.54923069332375312802315261249, −1.90932543245673322308318968095, −1.00002740608364711227614801236, −0.40019137138375085429916510160, 1.00951692317947935561092890706, 1.871204319441243711985783801620, 2.60152348181353895158523918415, 3.229759751467573625198617331006, 4.19417057015278449234563872181, 5.02326671361659138191317533426, 5.82803641142127242128177045720, 6.33827732078608311395464325365, 7.11350100398228076851207559413, 7.87336690215140747234498652715, 8.65283600647350561726037146023, 9.28999983958552104056357721494, 9.9985588374061298282580565278, 10.80716679233067358864977018259, 11.31035668446176144955341624268, 12.052562560319998092268833298302, 12.56781035365299583136341415783, 13.71712215725782458804459905456, 14.20575659078451044011585402311, 14.564706341502892994146169705946, 15.51774903182329659356497249185, 15.80047051329473929343070791857, 17.1070875378419617941131057148, 17.39535363611124060703536209749, 17.96268049714018490428802861791

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