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Properties

Label 1-1053-1053.734-r0-0-0
Degree $1$
Conductor $1053$
Sign $0.981 - 0.192i$
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  + (0.549 − 0.835i)2-s + (−0.396 − 0.918i)4-s + (−0.957 − 0.286i)5-s + (−0.802 + 0.597i)7-s + (−0.984 − 0.173i)8-s + (−0.766 + 0.642i)10-s + (−0.957 + 0.286i)11-s + (0.0581 + 0.998i)14-s + (−0.686 + 0.727i)16-s + (0.766 − 0.642i)17-s + (−0.984 − 0.173i)19-s + (0.116 + 0.993i)20-s + (−0.286 + 0.957i)22-s + (0.597 − 0.802i)23-s + (0.835 + 0.549i)25-s + ⋯
L(s)  = 1  + (0.549 − 0.835i)2-s + (−0.396 − 0.918i)4-s + (−0.957 − 0.286i)5-s + (−0.802 + 0.597i)7-s + (−0.984 − 0.173i)8-s + (−0.766 + 0.642i)10-s + (−0.957 + 0.286i)11-s + (0.0581 + 0.998i)14-s + (−0.686 + 0.727i)16-s + (0.766 − 0.642i)17-s + (−0.984 − 0.173i)19-s + (0.116 + 0.993i)20-s + (−0.286 + 0.957i)22-s + (0.597 − 0.802i)23-s + (0.835 + 0.549i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $0.981 - 0.192i$
Analytic conductor: \(4.89011\)
Root analytic conductor: \(4.89011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (734, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1053,\ (0:\ ),\ 0.981 - 0.192i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8501602070 - 0.08261965380i\)
\(L(\frac12)\) \(\approx\) \(0.8501602070 - 0.08261965380i\)
\(L(1)\) \(\approx\) \(0.8013285654 - 0.3566252511i\)
\(L(1)\) \(\approx\) \(0.8013285654 - 0.3566252511i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.549 - 0.835i)T \)
5 \( 1 + (-0.957 - 0.286i)T \)
7 \( 1 + (-0.802 + 0.597i)T \)
11 \( 1 + (-0.957 + 0.286i)T \)
17 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (-0.984 - 0.173i)T \)
23 \( 1 + (0.597 - 0.802i)T \)
29 \( 1 + (-0.893 + 0.448i)T \)
31 \( 1 + (-0.116 + 0.993i)T \)
37 \( 1 + (0.342 - 0.939i)T \)
41 \( 1 + (0.998 - 0.0581i)T \)
43 \( 1 + (0.286 + 0.957i)T \)
47 \( 1 + (0.116 + 0.993i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.957 + 0.286i)T \)
61 \( 1 + (-0.993 + 0.116i)T \)
67 \( 1 + (0.549 + 0.835i)T \)
71 \( 1 + (0.342 - 0.939i)T \)
73 \( 1 + (0.984 + 0.173i)T \)
79 \( 1 + (0.893 - 0.448i)T \)
83 \( 1 + (0.448 + 0.893i)T \)
89 \( 1 + (0.342 + 0.939i)T \)
97 \( 1 + (0.230 - 0.973i)T \)
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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

−21.66139090155935900029562482042, −20.94177678663414700210475521350, −20.02767426666599462133224700389, −18.97214736314405452176501282188, −18.6486462179858706312856375324, −17.28961527701319180989076326769, −16.712491454616879673512527525594, −16.000496184862574973281629522157, −15.2253880284761507911426639759, −14.75252970709974990934914681576, −13.56228098664183515090489687521, −13.03250225183294999186738118073, −12.28834701092439884432003017414, −11.2795989157821974268390426170, −10.43178714838197764971844547882, −9.38241035408030247422801710756, −8.209764130823375560782210583605, −7.726073973587378142295787783275, −6.92223273955046812322441231872, −6.06713612083372422187482431974, −5.16327735625799307214098585464, −3.95567805372525494475073502866, −3.593281945949247458709044490998, −2.53172685876508710556699186654, −0.39896768156687140718118601903, 0.85607535991650586271894155382, 2.39707842415118859663194528819, 3.02944986087642014627521679186, 3.995029734859520344444029429065, 4.89313358837641274823733030377, 5.64358542396974536651077831292, 6.7366300267606426809300244334, 7.77265571268495329101792862914, 8.86528465218709699849361684087, 9.47508811132910629144689115706, 10.606445406946717109556890826980, 11.111803492921467590775251082562, 12.36662113631249315414248288710, 12.496234182651596743604415252592, 13.26889880428925461809164995769, 14.49592720611303971775852085663, 15.111326623773617467465145108937, 15.91971842020472467922614234620, 16.52912176816654525246772736692, 17.98710466639866274537684494014, 18.676607998410855217204586783608, 19.287584647226436269387171797080, 19.90928188017042487059663586697, 20.84758732352621309956329961325, 21.29422320790291346241590935270

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