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Properties

Label 1-21e2-441.331-r0-0-0
Degree $1$
Conductor $441$
Sign $-0.944 + 0.328i$
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  + (−0.988 + 0.149i)2-s + (0.955 − 0.294i)4-s + (−0.900 + 0.433i)5-s + (−0.900 + 0.433i)8-s + (0.826 − 0.563i)10-s + (0.623 + 0.781i)11-s + (−0.988 + 0.149i)13-s + (0.826 − 0.563i)16-s + (0.955 + 0.294i)17-s + (−0.5 − 0.866i)19-s + (−0.733 + 0.680i)20-s + (−0.733 − 0.680i)22-s + (−0.222 + 0.974i)23-s + (0.623 − 0.781i)25-s + (0.955 − 0.294i)26-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)2-s + (0.955 − 0.294i)4-s + (−0.900 + 0.433i)5-s + (−0.900 + 0.433i)8-s + (0.826 − 0.563i)10-s + (0.623 + 0.781i)11-s + (−0.988 + 0.149i)13-s + (0.826 − 0.563i)16-s + (0.955 + 0.294i)17-s + (−0.5 − 0.866i)19-s + (−0.733 + 0.680i)20-s + (−0.733 − 0.680i)22-s + (−0.222 + 0.974i)23-s + (0.623 − 0.781i)25-s + (0.955 − 0.294i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.944 + 0.328i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ -0.944 + 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04745373722 + 0.2808066450i\)
\(L(\frac12)\) \(\approx\) \(0.04745373722 + 0.2808066450i\)
\(L(1)\) \(\approx\) \(0.4793672100 + 0.1414886859i\)
\(L(1)\) \(\approx\) \(0.4793672100 + 0.1414886859i\)

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 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (0.623 + 0.781i)T \)
13 \( 1 + (-0.988 + 0.149i)T \)
17 \( 1 + (0.955 + 0.294i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (-0.733 + 0.680i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.733 + 0.680i)T \)
41 \( 1 + (0.0747 - 0.997i)T \)
43 \( 1 + (0.0747 + 0.997i)T \)
47 \( 1 + (-0.988 + 0.149i)T \)
53 \( 1 + (-0.733 - 0.680i)T \)
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 \( 1 + (0.365 + 0.930i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.988 - 0.149i)T \)
89 \( 1 + (-0.988 - 0.149i)T \)
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

−23.93384051175900444816840880608, −22.87274523662188370726686668119, −21.79924768665563346610160966455, −20.813256975954006113397835828431, −20.111739449403670659637364928757, −19.1522056374930921083647067752, −18.86603567718612194156639824478, −17.56295402260594058130515538375, −16.560237406960438739092561219518, −16.34441454684644902637682857736, −15.07786975493916168167587421338, −14.327968504854425407139630939, −12.62832655919554475018487608138, −12.10326084203571000115793277870, −11.23360878850897224682914801709, −10.256698326265912341877886737770, −9.268687243522395488928862123572, −8.350242944332857498347307592922, −7.687565785447277092213182432541, −6.67190772744381331767052887980, −5.444790467485266746530466438080, −3.986828271384747757132914505498, −3.03742221114446672215858288217, −1.57795649631299633453736377182, −0.226478204665773083664332328433, 1.55259665310733025358229632797, 2.79484078170745703218910835128, 3.98771631239866435477099998805, 5.35688238122113427702908940281, 6.766212325978196415178353245313, 7.31668976401916484947857298870, 8.17007130484121293125444416270, 9.32862638148061156133185327166, 10.04837141705778431348841872732, 11.15283489574639719999021953476, 11.83475882594341428211833531399, 12.68901429423241352488416442951, 14.47501285253101099544680599777, 14.94586919054153881833653638651, 15.7957035778888622372166652570, 16.827719232594079790167632444905, 17.471759022665749604743579313023, 18.47719424957170675646578781823, 19.403609029052873289810553660351, 19.72805711676113773049760933261, 20.74423284196152302585996297119, 21.883779946292572194069449324865, 22.81473318849430039705114401067, 23.86605648353064851749288893398, 24.35760643228993523221782284334

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