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Properties

Label 1-1053-1053.437-r0-0-0
Degree $1$
Conductor $1053$
Sign $0.0743 - 0.997i$
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 + (−0.727 + 0.686i)5-s + (0.116 + 0.993i)7-s + (0.984 + 0.173i)8-s + (−0.173 − 0.984i)10-s + (0.957 − 0.286i)11-s + (−0.893 − 0.448i)14-s + (−0.686 + 0.727i)16-s + (−0.939 − 0.342i)17-s + (−0.342 − 0.939i)19-s + (0.918 + 0.396i)20-s + (−0.286 + 0.957i)22-s + (−0.993 − 0.116i)23-s + (0.0581 − 0.998i)25-s + ⋯
L(s)  = 1  + (−0.549 + 0.835i)2-s + (−0.396 − 0.918i)4-s + (−0.727 + 0.686i)5-s + (0.116 + 0.993i)7-s + (0.984 + 0.173i)8-s + (−0.173 − 0.984i)10-s + (0.957 − 0.286i)11-s + (−0.893 − 0.448i)14-s + (−0.686 + 0.727i)16-s + (−0.939 − 0.342i)17-s + (−0.342 − 0.939i)19-s + (0.918 + 0.396i)20-s + (−0.286 + 0.957i)22-s + (−0.993 − 0.116i)23-s + (0.0581 − 0.998i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0743 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0743 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1053\)    =    \(3^{4} \cdot 13\)
Sign: $0.0743 - 0.997i$
Analytic conductor: \(4.89011\)
Root analytic conductor: \(4.89011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1053} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1053,\ (0:\ ),\ 0.0743 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05205281801 - 0.04831455911i\)
\(L(\frac12)\) \(\approx\) \(0.05205281801 - 0.04831455911i\)
\(L(1)\) \(\approx\) \(0.5036563710 + 0.2657781071i\)
\(L(1)\) \(\approx\) \(0.5036563710 + 0.2657781071i\)

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 \( 1 + (-0.727 + 0.686i)T \)
7 \( 1 + (0.116 + 0.993i)T \)
11 \( 1 + (0.957 - 0.286i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (-0.993 - 0.116i)T \)
29 \( 1 + (-0.893 + 0.448i)T \)
31 \( 1 + (0.802 + 0.597i)T \)
37 \( 1 + (-0.642 - 0.766i)T \)
41 \( 1 + (0.549 + 0.835i)T \)
43 \( 1 + (-0.973 - 0.230i)T \)
47 \( 1 + (-0.802 + 0.597i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.957 - 0.286i)T \)
61 \( 1 + (0.396 - 0.918i)T \)
67 \( 1 + (-0.448 + 0.893i)T \)
71 \( 1 + (0.984 - 0.173i)T \)
73 \( 1 + (-0.984 - 0.173i)T \)
79 \( 1 + (-0.835 - 0.549i)T \)
83 \( 1 + (-0.549 + 0.835i)T \)
89 \( 1 + (0.984 + 0.173i)T \)
97 \( 1 + (-0.727 - 0.686i)T \)
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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

−21.45771456537997222185812089901, −20.64205125737251417664856769916, −20.0829045880507063195888096457, −19.58784123915464936908221988393, −18.832607576940876553783014645385, −17.75066599560881993405427581192, −17.017724182527330280422999061615, −16.62706604879509893649291490048, −15.61965121426561498083661123086, −14.54281493054377036953008302643, −13.53066280222915616361135723377, −12.922340820545108941028120736422, −11.89428738552223653350313217668, −11.5470143293660581300089164128, −10.48496549092977183468314132383, −9.79520984879455803688480642467, −8.837123621037714411933729808275, −8.124806175135320055447954979357, −7.409472949815781278000500457918, −6.39660236354069873108378549960, −4.81312213123925886730318482354, −3.99011676139415226313631246459, −3.67654760277791090460643179036, −2.01626065187981670646456811525, −1.22868795316472652546065097231, 0.03855814745008465990555668740, 1.69107424113091670759649178468, 2.81162442134658608273514505479, 4.07858076668493490655687420371, 4.93898954787326902636611431234, 6.12026500009444823664623913846, 6.64642865848303001459074975893, 7.50855312341907555169528239969, 8.5183539884280437172629648044, 8.9766724332141703751867004262, 9.94157228507991528456443278747, 11.08010754437670747814920947789, 11.507988575686474198123247495636, 12.59219063738267443708086630034, 13.81200854867663105293819760854, 14.49735167654159342476172052394, 15.24222276575982983920527442619, 15.74936504082069481713200258568, 16.518717001318757778678409600808, 17.63348994929229205207692435789, 18.105095466732332298946674369323, 18.9374172688796000844879532599, 19.55156877914132235166407452765, 20.15894882478100865251857277759, 21.71582494802515974026840576617

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