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Properties

Label 1-21e2-441.277-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  + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.826 + 0.563i)5-s + (−0.900 + 0.433i)8-s + (0.0747 + 0.997i)10-s + (−0.988 + 0.149i)11-s + (−0.988 + 0.149i)13-s + (−0.900 − 0.433i)16-s + (−0.733 + 0.680i)17-s + (−0.5 + 0.866i)19-s + (−0.733 + 0.680i)20-s + (−0.733 − 0.680i)22-s + (0.955 − 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.733 − 0.680i)26-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (0.826 + 0.563i)5-s + (−0.900 + 0.433i)8-s + (0.0747 + 0.997i)10-s + (−0.988 + 0.149i)11-s + (−0.988 + 0.149i)13-s + (−0.900 − 0.433i)16-s + (−0.733 + 0.680i)17-s + (−0.5 + 0.866i)19-s + (−0.733 + 0.680i)20-s + (−0.733 − 0.680i)22-s + (0.955 − 0.294i)23-s + (0.365 + 0.930i)25-s + (−0.733 − 0.680i)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: $\chi_{441} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ -0.944 + 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2568611865 + 1.519971497i\)
\(L(\frac12)\) \(\approx\) \(0.2568611865 + 1.519971497i\)
\(L(1)\) \(\approx\) \(0.9657011078 + 0.9193259349i\)
\(L(1)\) \(\approx\) \(0.9657011078 + 0.9193259349i\)

Euler product

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

−23.615066660787593034890470630906, −22.68041729041520777412865083887, −21.74159116233318514332901376244, −21.23099794447585696800721033, −20.373370096107063301641850938816, −19.64044767737490035746758527830, −18.595929703966086872494459088865, −17.75466712646207865642450212110, −16.86124674809868752667900241537, −15.57467361749436027469856086269, −14.87233399519977028886823843692, −13.533177913087440548693145114370, −13.31423370071516186078159721616, −12.33989631671728621145318702653, −11.31085372253131004621451598642, −10.37463008066565188221736834358, −9.55110164158728735911997152825, −8.74624479108413182495819005454, −7.22660616736902577306718111615, −5.999963284032514457915146001912, −5.08399652075847243744271665472, −4.471841594091577167349826585632, −2.77408361221633242419684309494, −2.23228413377176602446621527077, −0.66421362568321506005098655966, 2.15155888262478555351101593179, 2.99644828102715274609839498160, 4.38538645922241040754340803568, 5.3291431284523368274511745742, 6.229976056793195545681127088125, 7.08929473323729423515467798995, 8.017773253008257463574474713366, 9.14535137932682374475121368134, 10.20550264581467780460475533773, 11.1631685786304199659000870170, 12.57666313043892963750697572033, 13.05727995234631648242901259612, 14.10819930545464646813320600319, 14.81986537067332917351560095348, 15.51225294428939813820789961253, 16.75040664365739782372612097300, 17.323955021166926178909484248, 18.20552169437994174399837625086, 19.061634931592351235408902698983, 20.51345884601454204133937295846, 21.2980566897091688095543114375, 21.95920491562958404136391647946, 22.75647035002670958790730418935, 23.56819837492769775696758961934, 24.52629376355438605705032005840

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