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Properties

Label 1-21e2-441.398-r0-0-0
Degree $1$
Conductor $441$
Sign $-0.996 + 0.0782i$
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  + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (−0.988 + 0.149i)5-s + (−0.623 − 0.781i)8-s + (−0.623 + 0.781i)10-s + (0.733 − 0.680i)11-s + (−0.955 − 0.294i)13-s + (−0.988 − 0.149i)16-s + (−0.900 − 0.433i)17-s − 19-s + (0.0747 + 0.997i)20-s + (0.0747 − 0.997i)22-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.900 + 0.433i)26-s + ⋯
L(s)  = 1  + (0.733 − 0.680i)2-s + (0.0747 − 0.997i)4-s + (−0.988 + 0.149i)5-s + (−0.623 − 0.781i)8-s + (−0.623 + 0.781i)10-s + (0.733 − 0.680i)11-s + (−0.955 − 0.294i)13-s + (−0.988 − 0.149i)16-s + (−0.900 − 0.433i)17-s − 19-s + (0.0747 + 0.997i)20-s + (0.0747 − 0.997i)22-s + (−0.0747 + 0.997i)23-s + (0.955 − 0.294i)25-s + (−0.900 + 0.433i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.996 + 0.0782i$
Analytic conductor: \(2.04799\)
Root analytic conductor: \(2.04799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (398, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 441,\ (0:\ ),\ -0.996 + 0.0782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03457154632 - 0.8819064493i\)
\(L(\frac12)\) \(\approx\) \(-0.03457154632 - 0.8819064493i\)
\(L(1)\) \(\approx\) \(0.8221608924 - 0.6199276668i\)
\(L(1)\) \(\approx\) \(0.8221608924 - 0.6199276668i\)

Euler product

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

−24.34835167845613706799175402120, −23.71324003977183294511763937720, −22.82398860429534076162040596282, −22.19528216666518375844222690159, −21.30114927947134212474205888107, −20.11502963935278482535641147412, −19.63975165199580788367207650201, −18.360744378458940138819759868092, −17.18349437414167444446302253571, −16.702035724271689684859567314644, −15.59096132436770056209729254482, −14.939296077590149447838840314184, −14.29829313959160628911738003614, −12.980657970804111743691535269292, −12.28478463740081357720418818584, −11.63311492277782726347822452605, −10.41051600337021721792447643617, −8.86502777141937046315040463880, −8.28891429633803861823121289847, −6.93784310609502265599368643515, −6.69778865547664697502959952306, −4.972881565087609208105235552200, −4.412663434751213547269403711922, −3.44038540351986223094262314558, −2.080136146026300758004924116360, 0.36817634594610921510259057487, 2.03589089462226882740395477957, 3.19754616163147258252995175658, 4.07478262677494323501984155684, 4.93745550553191843895587498805, 6.210047239948996559977494550169, 7.12683772603174716683701062900, 8.38491279392805363777329329568, 9.4696099400939973206550862275, 10.53342182044400357409538959093, 11.52572138358554560888334777912, 11.89483178190931011451951244467, 13.03432965407245934728487706089, 13.868518454717570116611234419350, 14.97222917100570481974520216297, 15.37709244468767451177765198485, 16.541292751995898075510686323471, 17.66108808272848486492133788740, 18.958861479654784198238197782742, 19.42727267295481109914059913166, 20.077681475057843875485006031456, 21.11718294165871667798825388682, 22.064716256761875687271796877903, 22.613396627561987493681352305396, 23.498803959423969935267102497662

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