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Properties

Label 1-1053-1053.527-r0-0-0
Degree $1$
Conductor $1053$
Sign $-0.736 - 0.676i$
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  + (0.802 − 0.597i)2-s + (0.286 − 0.957i)4-s + (0.448 − 0.893i)5-s + (0.230 − 0.973i)7-s + (−0.342 − 0.939i)8-s + (−0.173 − 0.984i)10-s + (0.448 + 0.893i)11-s + (−0.396 − 0.918i)14-s + (−0.835 − 0.549i)16-s + (0.173 + 0.984i)17-s + (−0.342 − 0.939i)19-s + (−0.727 − 0.686i)20-s + (0.893 + 0.448i)22-s + (0.973 − 0.230i)23-s + (−0.597 − 0.802i)25-s + ⋯
L(s)  = 1  + (0.802 − 0.597i)2-s + (0.286 − 0.957i)4-s + (0.448 − 0.893i)5-s + (0.230 − 0.973i)7-s + (−0.342 − 0.939i)8-s + (−0.173 − 0.984i)10-s + (0.448 + 0.893i)11-s + (−0.396 − 0.918i)14-s + (−0.835 − 0.549i)16-s + (0.173 + 0.984i)17-s + (−0.342 − 0.939i)19-s + (−0.727 − 0.686i)20-s + (0.893 + 0.448i)22-s + (0.973 − 0.230i)23-s + (−0.597 − 0.802i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $-0.736 - 0.676i$
Analytic conductor: \(4.89011\)
Root analytic conductor: \(4.89011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1053,\ (0:\ ),\ -0.736 - 0.676i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9790659515 - 2.511901410i\)
\(L(\frac12)\) \(\approx\) \(0.9790659515 - 2.511901410i\)
\(L(1)\) \(\approx\) \(1.385138891 - 1.182399946i\)
\(L(1)\) \(\approx\) \(1.385138891 - 1.182399946i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.802 - 0.597i)T \)
5 \( 1 + (0.448 - 0.893i)T \)
7 \( 1 + (0.230 - 0.973i)T \)
11 \( 1 + (0.448 + 0.893i)T \)
17 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (0.973 - 0.230i)T \)
29 \( 1 + (0.993 - 0.116i)T \)
31 \( 1 + (0.727 - 0.686i)T \)
37 \( 1 + (-0.642 - 0.766i)T \)
41 \( 1 + (0.918 - 0.396i)T \)
43 \( 1 + (-0.893 + 0.448i)T \)
47 \( 1 + (-0.727 - 0.686i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.448 + 0.893i)T \)
61 \( 1 + (-0.686 + 0.727i)T \)
67 \( 1 + (0.802 + 0.597i)T \)
71 \( 1 + (-0.642 - 0.766i)T \)
73 \( 1 + (0.342 + 0.939i)T \)
79 \( 1 + (-0.993 + 0.116i)T \)
83 \( 1 + (0.116 + 0.993i)T \)
89 \( 1 + (-0.642 + 0.766i)T \)
97 \( 1 + (-0.998 - 0.0581i)T \)
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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

−21.66531067338668128570423475574, −21.49212928911119494748036118414, −20.64704651612040176928082825847, −19.30859563635234941444042478070, −18.6394279905738699463060271354, −17.84835515104551994641328906195, −17.07788015387117193051648108094, −16.14388286619786810400745380537, −15.461880073454907269500884071141, −14.58921770540827924226195132191, −14.145130235178116988539335613827, −13.381794435312755437058493601077, −12.32271244721817068283104274556, −11.62845888397070946184966918558, −10.936759188296409202267155565819, −9.74120781566093833722694973933, −8.7308693079846459572115101199, −8.02241892677912052230033431751, −6.86813154481890839768965379004, −6.298496472267303786632475276247, −5.50818497387619089087509834908, −4.7080567812914488529151565078, −3.25736101109060064961627061485, −2.94067481459664198358223634015, −1.71666003624758620610404992301, 0.887546274816796435285571007925, 1.6514821232340347034027790990, 2.70285352508473881318728403536, 4.0967157353681030794925785079, 4.46318699471230474258198378559, 5.35086774438372060662643738367, 6.41444290080486352319350798896, 7.16007739695325044749485639373, 8.43881886777758673114689070515, 9.39403292918626649883344189993, 10.168634155030225141756559688253, 10.85490295771922688620402435484, 11.865447899409406286453541870690, 12.63790155999632259366274278773, 13.24532988257995934160338177903, 13.930596947965371902451951758780, 14.80192251588081411110035465427, 15.51405005574973605631064780813, 16.64728784874085497260799030278, 17.27523925514427812120750020128, 18.00705133856539641756622221335, 19.484398819582723682941650458222, 19.68974387213581379083848694977, 20.60525207611455636034552325095, 21.15797321954953183803622072186

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