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Properties

Label 1-21e2-441.25-r0-0-0
Degree $1$
Conductor $441$
Sign $0.904 - 0.427i$
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  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.733 + 0.680i)5-s + (−0.222 − 0.974i)8-s + (0.955 − 0.294i)10-s + (0.826 − 0.563i)11-s + (0.826 − 0.563i)13-s + (−0.222 + 0.974i)16-s + (−0.988 − 0.149i)17-s + (−0.5 + 0.866i)19-s + (−0.988 − 0.149i)20-s + (−0.988 + 0.149i)22-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (−0.988 + 0.149i)26-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.733 + 0.680i)5-s + (−0.222 − 0.974i)8-s + (0.955 − 0.294i)10-s + (0.826 − 0.563i)11-s + (0.826 − 0.563i)13-s + (−0.222 + 0.974i)16-s + (−0.988 − 0.149i)17-s + (−0.5 + 0.866i)19-s + (−0.988 − 0.149i)20-s + (−0.988 + 0.149i)22-s + (0.365 − 0.930i)23-s + (0.0747 − 0.997i)25-s + (−0.988 + 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.904 - 0.427i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.904 - 0.427i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7461937261 - 0.1675197818i\)
\(L(\frac12)\) \(\approx\) \(0.7461937261 - 0.1675197818i\)
\(L(1)\) \(\approx\) \(0.6826866988 - 0.08396463852i\)
\(L(1)\) \(\approx\) \(0.6826866988 - 0.08396463852i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + (-0.733 + 0.680i)T \)
11 \( 1 + (0.826 - 0.563i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + (-0.988 - 0.149i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.365 - 0.930i)T \)
29 \( 1 + (-0.988 - 0.149i)T \)
31 \( 1 + T \)
37 \( 1 + (0.365 + 0.930i)T \)
41 \( 1 + (0.955 + 0.294i)T \)
43 \( 1 + (0.955 - 0.294i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.365 - 0.930i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (0.623 - 0.781i)T \)
67 \( 1 + T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (0.826 + 0.563i)T \)
79 \( 1 + T \)
83 \( 1 + (0.826 + 0.563i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
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

−24.25955916687519693560421323718, −23.51140762509313272880427403243, −22.69993159368239583397700189987, −21.300456723010709857099873268162, −20.42307212325562717455953452311, −19.57955254806342975578611357510, −19.16923196570187259226132815854, −17.86617827711820951470140117221, −17.251525396171545696146312675424, −16.33340049383901817097873721558, −15.58124514435742529432290871226, −14.910773086513163480697297511920, −13.686517236289514886879373013150, −12.56004268701278305367140547767, −11.397764367888356462759485925011, −10.98405044741180068138315775414, −9.3045962791031241876971261031, −9.07741352091457967436457550403, −7.9687404789160436954468413664, −7.04242723821933461775962126568, −6.17091861054590914809254752395, −4.83909767988714778243901080418, −3.87788325778029631384198518778, −2.11573906758137473432325225783, −0.961195760347226971852115396949, 0.80693918545159013279144303091, 2.31224757347518183487197666256, 3.41458352601863135411222634675, 4.18405600374474704847600543655, 6.187131326298146053906869994867, 6.83724052414532267272627035270, 8.080624514975977023759575247091, 8.59165033457922906006185428199, 9.78625966391850102814899751315, 10.87414389868667176024163503931, 11.2702885985506173316037900962, 12.26674884375606398524316662511, 13.28831814126106626092997862183, 14.570939682141952484090774819894, 15.47535700577123107860230106328, 16.29360560504775067672746869583, 17.16898007445304216244923224965, 18.20462176296968633038139390864, 18.82794702984745124708504300414, 19.568758663851471933545722210, 20.37644406793602078402003526973, 21.240618798219498498156681810597, 22.37019462440605052565277048106, 22.8463598660474257545742362996, 24.23403683776666348264390292522

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