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Properties

Label 1-21e2-441.101-r0-0-0
Degree $1$
Conductor $441$
Sign $0.250 - 0.968i$
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  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.988 − 0.149i)5-s + (−0.623 + 0.781i)8-s + (−0.365 + 0.930i)10-s + (0.733 + 0.680i)11-s + (0.733 + 0.680i)13-s + (0.623 + 0.781i)16-s + (0.826 + 0.563i)17-s + (0.5 − 0.866i)19-s + (0.826 + 0.563i)20-s + (0.826 − 0.563i)22-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (0.826 − 0.563i)26-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.988 − 0.149i)5-s + (−0.623 + 0.781i)8-s + (−0.365 + 0.930i)10-s + (0.733 + 0.680i)11-s + (0.733 + 0.680i)13-s + (0.623 + 0.781i)16-s + (0.826 + 0.563i)17-s + (0.5 − 0.866i)19-s + (0.826 + 0.563i)20-s + (0.826 − 0.563i)22-s + (−0.0747 − 0.997i)23-s + (0.955 + 0.294i)25-s + (0.826 − 0.563i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.250 - 0.968i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ 0.250 - 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9238635931 - 0.7154632203i\)
\(L(\frac12)\) \(\approx\) \(0.9238635931 - 0.7154632203i\)
\(L(1)\) \(\approx\) \(0.8838012861 - 0.4699675748i\)
\(L(1)\) \(\approx\) \(0.8838012861 - 0.4699675748i\)

Euler product

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

−24.17388315808419297627373965426, −23.427361955144572530492355614299, −22.726200714161571703455008725680, −22.0436455659723744799301739616, −20.88480585606454429172027934559, −19.89173304497479259287014016151, −18.777512612063110104157155333422, −18.30260095062198325605177984855, −17.031363891263621967064132344029, −16.27948761745965266686385294044, −15.65269464825688625701414155570, −14.67970749100019686714970846544, −14.005840487229129261250049075728, −12.92906632305718089450944992643, −11.98127333984836725624466799675, −11.11853165560465031226076749751, −9.74435235188700366841958031475, −8.70575244972599493008218032868, −7.876100373090598841381041644780, −7.1667672283901441192132314003, −5.98460871221080466931731189066, −5.1798316525878836144292162259, −3.66829407528347224126486330693, −3.446959001663798576289080905788, −0.97137204705306009824985418367, 0.92239033776487381418482629624, 2.17297389242250859550146480589, 3.621064892703081909844924661719, 4.12568778955047580555430913411, 5.22801750138660285689428100522, 6.572330540829049654613152677055, 7.79682379149293185356835828531, 8.85753297636902562571236715413, 9.590369126505404546999044350488, 10.86422258387585840776690230899, 11.48805354266598773244526959523, 12.33395961101203788834775176901, 13.037328587737611376809615884027, 14.29752999172196312983399329141, 14.88072233591268483598546922818, 16.041181971964041441691323243549, 17.021810243264649847223764172412, 18.13837988500297150567027979813, 18.95369302486208944189695819229, 19.66511076208657472469588213642, 20.39892310758937542317718264796, 21.15939539026515411724352434421, 22.2540043145126963029503369863, 22.87739300221050384714951428983, 23.70248364261093443396377247505

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