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Properties

Label 1-21e2-441.265-r1-0-0
Degree $1$
Conductor $441$
Sign $0.0142 - 0.999i$
Analytic cond. $47.3920$
Root an. cond. $47.3920$
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  + (−0.988 + 0.149i)2-s + (0.955 − 0.294i)4-s + (−0.826 − 0.563i)5-s + (−0.900 + 0.433i)8-s + (0.900 + 0.433i)10-s + (−0.988 + 0.149i)11-s + (−0.365 + 0.930i)13-s + (0.826 − 0.563i)16-s + (0.222 + 0.974i)17-s − 19-s + (−0.955 − 0.294i)20-s + (0.955 − 0.294i)22-s + (0.955 − 0.294i)23-s + (0.365 + 0.930i)25-s + (0.222 − 0.974i)26-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)2-s + (0.955 − 0.294i)4-s + (−0.826 − 0.563i)5-s + (−0.900 + 0.433i)8-s + (0.900 + 0.433i)10-s + (−0.988 + 0.149i)11-s + (−0.365 + 0.930i)13-s + (0.826 − 0.563i)16-s + (0.222 + 0.974i)17-s − 19-s + (−0.955 − 0.294i)20-s + (0.955 − 0.294i)22-s + (0.955 − 0.294i)23-s + (0.365 + 0.930i)25-s + (0.222 − 0.974i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0142 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0142 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.0142 - 0.999i$
Analytic conductor: \(47.3920\)
Root analytic conductor: \(47.3920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (1:\ ),\ 0.0142 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2997197364 - 0.2954795880i\)
\(L(\frac12)\) \(\approx\) \(0.2997197364 - 0.2954795880i\)
\(L(1)\) \(\approx\) \(0.5172622137 + 0.001847411597i\)
\(L(1)\) \(\approx\) \(0.5172622137 + 0.001847411597i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.988 + 0.149i)T \)
5 \( 1 + (-0.826 - 0.563i)T \)
11 \( 1 + (-0.988 + 0.149i)T \)
13 \( 1 + (-0.365 + 0.930i)T \)
17 \( 1 + (0.222 + 0.974i)T \)
19 \( 1 - T \)
23 \( 1 + (0.955 - 0.294i)T \)
29 \( 1 + (0.955 + 0.294i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + (-0.826 - 0.563i)T \)
43 \( 1 + (0.826 - 0.563i)T \)
47 \( 1 + (0.988 - 0.149i)T \)
53 \( 1 + (-0.222 + 0.974i)T \)
59 \( 1 + (-0.0747 - 0.997i)T \)
61 \( 1 + (-0.955 - 0.294i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.365 - 0.930i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
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

−24.11281534344613613421571498546, −23.26673120249826779069776315902, −22.38796206101429707327450761023, −21.190590884197535374369151082493, −20.51525651714485864802707068166, −19.53052840864715280200808351134, −18.9020135522007744489819240731, −18.15858038045080446275338848689, −17.28505652289276866227150880900, −16.25851503686679560620853260661, −15.41874440455429513128699640892, −14.9274904256763926087575334467, −13.385909301884794980634085706527, −12.32571923374344499392888324387, −11.475645738259162489522609537872, −10.61184935701746305951067864143, −9.9824072658000354533435182007, −8.66250777134288183418932579481, −7.84988332248766388145648201424, −7.21054838357165321452407727576, −6.08278489436080532362117697792, −4.68408290787632751376529527263, −3.11661461788742241364958670111, −2.5954270130612488112735256071, −0.7773603460317564261882003491, 0.22134661511772876877087437620, 1.561119489500151100777992972495, 2.79450522314007117080385749390, 4.23217466052016976990596994996, 5.34355267915709891797743007419, 6.656649348893765752227654393459, 7.50224265352645694315725165136, 8.46029128324716141636794928389, 9.01025003868387592557596390691, 10.35420820933394059027070624123, 10.9435013550706053132947234885, 12.1537405793486884043748339345, 12.68980117952928658093405080760, 14.245707503048020889298283703507, 15.30599493881192884266584345359, 15.843544470596801475214173070214, 16.84604788127255123058243309328, 17.38198021846323368536067453690, 18.70999763597509485171090750043, 19.17607382439244646191866347682, 19.97561122445620986175121999836, 20.95812963523736169246983607380, 21.535441219289251846713670037102, 23.32617585228545606303162964088, 23.61869064693047182390014807511

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