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Properties

Label 1-21e2-441.383-r0-0-0
Degree $1$
Conductor $441$
Sign $0.577 + 0.816i$
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 + (0.955 + 0.294i)5-s + (0.222 + 0.974i)8-s + (0.733 + 0.680i)10-s + (−0.0747 − 0.997i)11-s + (−0.0747 − 0.997i)13-s + (−0.222 + 0.974i)16-s + (0.365 + 0.930i)17-s + (0.5 + 0.866i)19-s + (0.365 + 0.930i)20-s + (0.365 − 0.930i)22-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (0.365 − 0.930i)26-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (0.955 + 0.294i)5-s + (0.222 + 0.974i)8-s + (0.733 + 0.680i)10-s + (−0.0747 − 0.997i)11-s + (−0.0747 − 0.997i)13-s + (−0.222 + 0.974i)16-s + (0.365 + 0.930i)17-s + (0.5 + 0.866i)19-s + (0.365 + 0.930i)20-s + (0.365 − 0.930i)22-s + (0.988 − 0.149i)23-s + (0.826 + 0.563i)25-s + (0.365 − 0.930i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.407296198 + 1.244996797i\)
\(L(\frac12)\) \(\approx\) \(2.407296198 + 1.244996797i\)
\(L(1)\) \(\approx\) \(1.909287932 + 0.6530995356i\)
\(L(1)\) \(\approx\) \(1.909287932 + 0.6530995356i\)

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 \( 1 + (0.955 + 0.294i)T \)
11 \( 1 + (-0.0747 - 0.997i)T \)
13 \( 1 + (-0.0747 - 0.997i)T \)
17 \( 1 + (0.365 + 0.930i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.988 - 0.149i)T \)
29 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.988 - 0.149i)T \)
41 \( 1 + (-0.733 + 0.680i)T \)
43 \( 1 + (-0.733 - 0.680i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.988 - 0.149i)T \)
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 \( 1 + (-0.0747 + 0.997i)T \)
79 \( 1 + T \)
83 \( 1 + (0.0747 - 0.997i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
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

−23.86079859873973275247381963085, −22.97112771542080781603985927131, −22.114506843950860363420682001643, −21.41418126736457118380060295494, −20.61144054162358287140297128587, −20.01453904512028900354137103299, −18.82952839273458300633944522010, −17.99579382457411062208824727498, −16.89089057130109435807840594269, −16.02749344444942749295166155993, −14.93619766308044304193992007720, −14.15163026424122366289959258882, −13.391763190151510383794335380541, −12.595721698853974768155487126398, −11.703565250826004509036094795823, −10.727521187353419709880768884085, −9.639464297757658939832510166410, −9.14303243038648580074205584606, −7.199557640811659879742221298393, −6.62788777632278605947074819243, −5.15003398079827113816645578579, −4.91403969373661533796144579118, −3.40835548727573050000940338245, −2.266901866248837632976255455, −1.39431793627669891481394660140, 1.644762259830150439793728884571, 2.94744311399892892021474209425, 3.67763425357773002213982115748, 5.3131585724620302979500841878, 5.72439282908864387013823118017, 6.70443790840168692933981197624, 7.8325676145123672537100740758, 8.74426103260588216450866008035, 10.16784889124079508340905720575, 10.90146571391144667152901663937, 12.05005901368999197237961198276, 13.1024684372171438556806937592, 13.57851266322657097385232941674, 14.62653385355579349505324477005, 15.18397749981015933325457867149, 16.4740332081768169325827052574, 17.00525893314402035423365907979, 17.991138990289497743600141334717, 18.97373369464291309151766184104, 20.21280818205195511780735572475, 21.101474272613346279415846264731, 21.65084523884068112009305372407, 22.520935732049443483712092449264, 23.188236739783129740691703026813, 24.35585095388887119433755172330

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