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Properties

Label 1-21e2-441.38-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: $\chi_{441} (38, \cdot )$
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

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

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