Properties

Label 1-21e2-441.331-r0-0-0
Degree 11
Conductor 441441
Sign 0.944+0.328i-0.944 + 0.328i
Analytic cond. 2.047992.04799
Root an. cond. 2.047992.04799
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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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

Λ(s)=(441s/2ΓR(s)L(s)=((0.944+0.328i)Λ(1s)\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}
Λ(s)=(441s/2ΓR(s)L(s)=((0.944+0.328i)Λ(1s)\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: 11
Conductor: 441441    =    32723^{2} \cdot 7^{2}
Sign: 0.944+0.328i-0.944 + 0.328i
Analytic conductor: 2.047992.04799
Root analytic conductor: 2.047992.04799
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ441(331,)\chi_{441} (331, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 441, (0: ), 0.944+0.328i)(1,\ 441,\ (0:\ ),\ -0.944 + 0.328i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.04745373722+0.2808066450i0.04745373722 + 0.2808066450i
L(12)L(\frac12) \approx 0.04745373722+0.2808066450i0.04745373722 + 0.2808066450i
L(1)L(1) \approx 0.4793672100+0.1414886859i0.4793672100 + 0.1414886859i
L(1)L(1) \approx 0.4793672100+0.1414886859i0.4793672100 + 0.1414886859i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
7 1 1
good2 1+(0.988+0.149i)T 1 + (-0.988 + 0.149i)T
5 1+(0.900+0.433i)T 1 + (-0.900 + 0.433i)T
11 1+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
13 1+(0.988+0.149i)T 1 + (-0.988 + 0.149i)T
17 1+(0.955+0.294i)T 1 + (0.955 + 0.294i)T
19 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
23 1+(0.222+0.974i)T 1 + (-0.222 + 0.974i)T
29 1+(0.733+0.680i)T 1 + (-0.733 + 0.680i)T
31 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
37 1+(0.733+0.680i)T 1 + (-0.733 + 0.680i)T
41 1+(0.07470.997i)T 1 + (0.0747 - 0.997i)T
43 1+(0.0747+0.997i)T 1 + (0.0747 + 0.997i)T
47 1+(0.988+0.149i)T 1 + (-0.988 + 0.149i)T
53 1+(0.7330.680i)T 1 + (-0.733 - 0.680i)T
59 1+(0.0747+0.997i)T 1 + (0.0747 + 0.997i)T
61 1+(0.955+0.294i)T 1 + (0.955 + 0.294i)T
67 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
71 1+(0.222+0.974i)T 1 + (-0.222 + 0.974i)T
73 1+(0.365+0.930i)T 1 + (0.365 + 0.930i)T
79 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
83 1+(0.9880.149i)T 1 + (-0.988 - 0.149i)T
89 1+(0.9880.149i)T 1 + (-0.988 - 0.149i)T
97 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
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   L(s)=p (1αpps)1L(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 ZZ-function along the critical line