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Properties

Label 1-1053-1053.200-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: $\chi_{1053} (200, \cdot )$
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

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

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