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Properties

Label 1-1053-1053.1024-r0-0-0
Degree $1$
Conductor $1053$
Sign $0.950 + 0.311i$
Analytic cond. $4.89011$
Root an. cond. $4.89011$
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  + (0.993 + 0.116i)2-s + (0.973 + 0.230i)4-s + (0.0581 − 0.998i)5-s + (0.686 + 0.727i)7-s + (0.939 + 0.342i)8-s + (0.173 − 0.984i)10-s + (0.0581 + 0.998i)11-s + (0.597 + 0.802i)14-s + (0.893 + 0.448i)16-s + (0.173 − 0.984i)17-s + (0.939 + 0.342i)19-s + (0.286 − 0.957i)20-s + (−0.0581 + 0.998i)22-s + (−0.686 + 0.727i)23-s + (−0.993 − 0.116i)25-s + ⋯
L(s)  = 1  + (0.993 + 0.116i)2-s + (0.973 + 0.230i)4-s + (0.0581 − 0.998i)5-s + (0.686 + 0.727i)7-s + (0.939 + 0.342i)8-s + (0.173 − 0.984i)10-s + (0.0581 + 0.998i)11-s + (0.597 + 0.802i)14-s + (0.893 + 0.448i)16-s + (0.173 − 0.984i)17-s + (0.939 + 0.342i)19-s + (0.286 − 0.957i)20-s + (−0.0581 + 0.998i)22-s + (−0.686 + 0.727i)23-s + (−0.993 − 0.116i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $0.950 + 0.311i$
Analytic conductor: \(4.89011\)
Root analytic conductor: \(4.89011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (1024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1053,\ (0:\ ),\ 0.950 + 0.311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.330226033 + 0.5320491021i\)
\(L(\frac12)\) \(\approx\) \(3.330226033 + 0.5320491021i\)
\(L(1)\) \(\approx\) \(2.169518684 + 0.1690739609i\)
\(L(1)\) \(\approx\) \(2.169518684 + 0.1690739609i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.993 + 0.116i)T \)
5 \( 1 + (0.0581 - 0.998i)T \)
7 \( 1 + (0.686 + 0.727i)T \)
11 \( 1 + (0.0581 + 0.998i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (-0.686 + 0.727i)T \)
29 \( 1 + (0.396 + 0.918i)T \)
31 \( 1 + (0.286 + 0.957i)T \)
37 \( 1 + (-0.766 - 0.642i)T \)
41 \( 1 + (-0.597 - 0.802i)T \)
43 \( 1 + (-0.0581 - 0.998i)T \)
47 \( 1 + (0.286 - 0.957i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.0581 - 0.998i)T \)
61 \( 1 + (-0.286 + 0.957i)T \)
67 \( 1 + (0.993 - 0.116i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (0.396 + 0.918i)T \)
83 \( 1 + (-0.396 - 0.918i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.835 - 0.549i)T \)
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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

−21.54199242488135069271244394772, −20.93339120318355370095241216192, −20.03883519673256767549623762683, −19.28428452169408271332088859713, −18.53294381204506640882876341873, −17.49746553318752888491197117916, −16.71539113536115668006077390810, −15.78342145296490184299219871417, −14.995975640662543134061334729727, −14.213841579272380189501140617378, −13.82057366153870081620675600462, −13.00339795796499169343402094308, −11.72263503215561299264029903352, −11.32476819882085423198139803538, −10.502834693766455028245976143101, −9.88290048595957592876423374869, −8.18192062503619989094085260090, −7.63957875377139989352502782467, −6.5121747540027064802887783502, −6.05075752119140288812146991714, −4.90733080606376830135570573582, −3.95951465118165159056242338981, −3.24787957554500245581538174447, −2.28908866731653117198052511241, −1.147139595784508027552580995183, 1.42148016161094724836754139009, 2.113659252746066377677444537340, 3.33269857942684287085430412058, 4.38862494680330882166569754203, 5.280317610272232351916272157860, 5.44884860479800485951206080834, 6.90558782828392619800087788565, 7.64526002673299148433872766123, 8.57325738919258168388506187991, 9.48545487008924911393567353157, 10.49205071949584022344758969825, 11.78800183615814620298170457144, 12.04474787194609946475081638161, 12.71376812685722443649189888211, 13.90194302268637831724070396223, 14.20466525654606707962153628601, 15.54426025957852592543985784813, 15.69307846313969720714885921347, 16.72345988050784191646164864430, 17.58440193699660007692817410195, 18.28953061921969787398685081667, 19.59801678266371001670439844575, 20.356185086941471862527747345752, 20.751863801047020139773880314821, 21.59388654080498869986925563276

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