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Properties

Label 1-21e2-441.358-r0-0-0
Degree $1$
Conductor $441$
Sign $-0.0818 + 0.996i$
Analytic cond. $2.04799$
Root an. cond. $2.04799$
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  + (0.0747 + 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.955 + 0.294i)5-s + (−0.222 − 0.974i)8-s + (−0.222 + 0.974i)10-s + (0.0747 + 0.997i)11-s + (0.826 − 0.563i)13-s + (0.955 − 0.294i)16-s + (0.623 − 0.781i)17-s + 19-s + (−0.988 − 0.149i)20-s + (−0.988 + 0.149i)22-s + (−0.988 + 0.149i)23-s + (0.826 + 0.563i)25-s + (0.623 + 0.781i)26-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.955 + 0.294i)5-s + (−0.222 − 0.974i)8-s + (−0.222 + 0.974i)10-s + (0.0747 + 0.997i)11-s + (0.826 − 0.563i)13-s + (0.955 − 0.294i)16-s + (0.623 − 0.781i)17-s + 19-s + (−0.988 − 0.149i)20-s + (−0.988 + 0.149i)22-s + (−0.988 + 0.149i)23-s + (0.826 + 0.563i)25-s + (0.623 + 0.781i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0818 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0818 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.0818 + 0.996i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (358, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ -0.0818 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.044850180 + 1.134156176i\)
\(L(\frac12)\) \(\approx\) \(1.044850180 + 1.134156176i\)
\(L(1)\) \(\approx\) \(1.034585823 + 0.6635340569i\)
\(L(1)\) \(\approx\) \(1.034585823 + 0.6635340569i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.0747 + 0.997i)T \)
5 \( 1 + (0.955 + 0.294i)T \)
11 \( 1 + (0.0747 + 0.997i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.988 + 0.149i)T \)
29 \( 1 + (-0.988 - 0.149i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 + (0.955 + 0.294i)T \)
43 \( 1 + (0.955 - 0.294i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + (-0.733 - 0.680i)T \)
61 \( 1 + (-0.988 - 0.149i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.826 + 0.563i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
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

−23.91118346814191900403057941656, −22.670232522896727433722608298547, −21.90028753901106586104134346273, −21.24063163348037249773535547469, −20.57891081301345456417041594999, −19.64146320407586444879341319764, −18.57330458687203508021102831358, −18.1564124137898629375782192704, −16.95464057851723514114182845873, −16.32177531648045875717364162653, −14.74594083299344223116717432323, −13.82918457838418773833355499973, −13.38736769677475652069096464433, −12.35118979143968587161345746967, −11.38207026543469402618483484021, −10.559896119173517639357159228770, −9.58720792371701964940285511681, −8.90313368628777682017328656507, −7.89913537006361748710887232576, −6.056664733641609914419383179372, −5.5978134097689816700184692691, −4.202324269427225437739763604179, −3.26388851614859804396665324684, −2.004836502300683341406920648160, −1.06492209074844404745322014414, 1.32529244594247932154484930285, 2.898966488457966548652313233862, 4.146570593247758337124943874006, 5.416415653219369339872276183964, 5.9305539921593317360933436378, 7.13551521676553274192093737506, 7.789959468801205519417098406535, 9.21555232645302900509966110552, 9.68109603111311823384096032178, 10.759490913706423756920051530316, 12.246902490278545728727525219540, 13.11417365612161293444738955556, 14.01117943322367926835095223306, 14.571379488293897493718947501477, 15.680517174958291279471422798088, 16.36977601757386902350135460215, 17.50500794252145378884069693152, 18.0217114265996226480102745200, 18.6387785076223246546002595674, 20.12586486936221100833359473074, 20.97467070695984098926063571013, 22.00464118579321514582924333398, 22.70391741746116540371768170984, 23.33751618857069901144364287392, 24.547325149031644445485172623891

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