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Properties

Label 1-2e8-256.173-r0-0-0
Degree $1$
Conductor $256$
Sign $0.999 + 0.0245i$
Analytic cond. $1.18885$
Root an. cond. $1.18885$
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  + (−0.471 + 0.881i)3-s + (−0.773 − 0.634i)5-s + (0.831 + 0.555i)7-s + (−0.555 − 0.831i)9-s + (0.290 − 0.956i)11-s + (−0.634 − 0.773i)13-s + (0.923 − 0.382i)15-s + (0.923 + 0.382i)17-s + (0.995 + 0.0980i)19-s + (−0.881 + 0.471i)21-s + (−0.980 − 0.195i)23-s + (0.195 + 0.980i)25-s + (0.995 − 0.0980i)27-s + (0.956 − 0.290i)29-s + (0.707 − 0.707i)31-s + ⋯
L(s)  = 1  + (−0.471 + 0.881i)3-s + (−0.773 − 0.634i)5-s + (0.831 + 0.555i)7-s + (−0.555 − 0.831i)9-s + (0.290 − 0.956i)11-s + (−0.634 − 0.773i)13-s + (0.923 − 0.382i)15-s + (0.923 + 0.382i)17-s + (0.995 + 0.0980i)19-s + (−0.881 + 0.471i)21-s + (−0.980 − 0.195i)23-s + (0.195 + 0.980i)25-s + (0.995 − 0.0980i)27-s + (0.956 − 0.290i)29-s + (0.707 − 0.707i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(256\)    =    \(2^{8}\)
Sign: $0.999 + 0.0245i$
Analytic conductor: \(1.18885\)
Root analytic conductor: \(1.18885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{256} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 256,\ (0:\ ),\ 0.999 + 0.0245i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9632145303 + 0.01182101408i\)
\(L(\frac12)\) \(\approx\) \(0.9632145303 + 0.01182101408i\)
\(L(1)\) \(\approx\) \(0.8944049568 + 0.06952593539i\)
\(L(1)\) \(\approx\) \(0.8944049568 + 0.06952593539i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-0.471 + 0.881i)T \)
5 \( 1 + (-0.773 - 0.634i)T \)
7 \( 1 + (0.831 + 0.555i)T \)
11 \( 1 + (0.290 - 0.956i)T \)
13 \( 1 + (-0.634 - 0.773i)T \)
17 \( 1 + (0.923 + 0.382i)T \)
19 \( 1 + (0.995 + 0.0980i)T \)
23 \( 1 + (-0.980 - 0.195i)T \)
29 \( 1 + (0.956 - 0.290i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (0.0980 + 0.995i)T \)
41 \( 1 + (0.195 - 0.980i)T \)
43 \( 1 + (0.471 + 0.881i)T \)
47 \( 1 + (0.382 - 0.923i)T \)
53 \( 1 + (0.956 + 0.290i)T \)
59 \( 1 + (-0.634 + 0.773i)T \)
61 \( 1 + (-0.881 - 0.471i)T \)
67 \( 1 + (0.881 + 0.471i)T \)
71 \( 1 + (0.555 - 0.831i)T \)
73 \( 1 + (0.831 - 0.555i)T \)
79 \( 1 + (-0.382 - 0.923i)T \)
83 \( 1 + (0.0980 - 0.995i)T \)
89 \( 1 + (-0.980 + 0.195i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−25.9555771967874219287348666498, −24.83597305174840891050104118618, −23.99287574173868152382969520034, −23.25519781629981015795511665263, −22.631017963080932560426111709467, −21.49444062158545160675148445558, −20.097593899709212170521273342603, −19.53793722619941943129877946574, −18.39276968531884300193818707693, −17.79281984355883828033027455343, −16.817430806085189523253992856650, −15.73735640758907196348333656951, −14.24925827560000692329452564316, −14.11278313603688704088082340930, −12.282945798674880265817361824340, −11.897600986993054293233950508699, −10.9271089132350374964301202209, −9.80888421938883563674877444329, −8.093428273112737376973713636829, −7.38698328243918670099071175106, −6.74117700433912133563874101337, −5.19346905557356118402920297713, −4.14333922965210292504607952635, −2.544978457771775444462505221774, −1.21160040138412631365877914872, 0.90078026908181806014845677970, 3.0271578818661585025821332977, 4.168403482235240404502636456982, 5.179796087915701568709040975523, 5.93172445089577752908159843477, 7.82462547018797338516798128506, 8.496567745306269996570705145562, 9.660461630321521827554773664078, 10.742471898459249890668810204950, 11.9025410374617395221713705420, 12.11600743829818498246819046761, 13.90002624329216362180019024530, 14.95961641593014302885770301041, 15.69102156150967158094389116026, 16.57554697101813117140248649173, 17.37164802260458803872324557454, 18.485673656646945859889958354945, 19.67930759220220669496704899668, 20.56124960585482435305972332250, 21.37780584719584589072145852841, 22.20769452511513209671865365312, 23.149787739743643189975918765703, 24.19551548261621309326990899057, 24.736819925772197627403244439622, 26.204498361877300504895917323531

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