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Properties

Label 1-21e2-441.419-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  + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (0.0747 + 0.997i)5-s + (0.900 + 0.433i)8-s + (0.900 − 0.433i)10-s + (−0.365 − 0.930i)11-s + (0.988 + 0.149i)13-s + (0.0747 − 0.997i)16-s + (−0.222 + 0.974i)17-s − 19-s + (−0.733 − 0.680i)20-s + (−0.733 + 0.680i)22-s + (0.733 − 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.222 − 0.974i)26-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)2-s + (−0.733 + 0.680i)4-s + (0.0747 + 0.997i)5-s + (0.900 + 0.433i)8-s + (0.900 − 0.433i)10-s + (−0.365 − 0.930i)11-s + (0.988 + 0.149i)13-s + (0.0747 − 0.997i)16-s + (−0.222 + 0.974i)17-s − 19-s + (−0.733 − 0.680i)20-s + (−0.733 + 0.680i)22-s + (0.733 − 0.680i)23-s + (−0.988 + 0.149i)25-s + (−0.222 − 0.974i)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: $\chi_{441} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.944 + 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9278650385 + 0.1568006473i\)
\(L(\frac12)\) \(\approx\) \(0.9278650385 + 0.1568006473i\)
\(L(1)\) \(\approx\) \(0.8370002046 - 0.09919975920i\)
\(L(1)\) \(\approx\) \(0.8370002046 - 0.09919975920i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.365 - 0.930i)T \)
5 \( 1 + (0.0747 + 0.997i)T \)
11 \( 1 + (-0.365 - 0.930i)T \)
13 \( 1 + (0.988 + 0.149i)T \)
17 \( 1 + (-0.222 + 0.974i)T \)
19 \( 1 - T \)
23 \( 1 + (0.733 - 0.680i)T \)
29 \( 1 + (0.733 + 0.680i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.222 + 0.974i)T \)
41 \( 1 + (0.0747 + 0.997i)T \)
43 \( 1 + (0.0747 - 0.997i)T \)
47 \( 1 + (0.365 + 0.930i)T \)
53 \( 1 + (0.222 + 0.974i)T \)
59 \( 1 + (0.826 + 0.563i)T \)
61 \( 1 + (0.733 + 0.680i)T \)
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 \( 1 + (-0.988 + 0.149i)T \)
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

−24.092370433511277539939371839259, −23.24094660302263284960975637430, −22.78660997654412529144109848232, −21.30445688363538540766209068768, −20.589979189290647786347284238614, −19.61530821888004218321885645018, −18.68853193501032440890981844766, −17.64505743151604467048698020250, −17.21639576752342766093638751823, −16.01475117559387342835571596090, −15.65464812089106073619128191111, −14.5797784704132157706937006946, −13.38853712258524400300169154209, −12.97831382929760204123604220527, −11.65000371371048894787173167702, −10.38365976276807704408866565225, −9.45810688944293215164774622682, −8.69645427408161127263863804723, −7.84774332901957858960226755850, −6.838507822418131671381813052361, −5.745159384969690346506782093493, −4.87797464148859236527855830797, −4.02424685409506355013373458791, −2.04874664996025019274967039562, −0.69087000721423723737423599198, 1.28124289699162297780870383510, 2.60767646062097181889935991014, 3.37460624751299936793585538965, 4.42234677999316232178420747756, 5.94121227016177442517798906118, 6.89862026982805827743890285719, 8.30990289691671592187134903876, 8.75716894517234497294158349723, 10.28381115258722280120948843232, 10.70221198705578565960570009453, 11.4272925611479835183479350912, 12.61699603570859685603496232289, 13.48703759130407245437077266159, 14.25774574160321638468362948085, 15.361381416091199327440749576220, 16.5004335810797305253356751044, 17.44661568654582103521963296788, 18.33409274689386971561249092670, 18.965863333711237955977989302421, 19.57319677911819699814025117298, 20.83874889482827464268114641081, 21.468305918787816964919650420216, 22.13045844019936140182906239525, 23.14228893948273583053489131355, 23.77203879844365058711860646158

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