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Properties

Label 1-511-511.229-r0-0-0
Degree $1$
Conductor $511$
Sign $0.284 + 0.958i$
Analytic cond. $2.37307$
Root an. cond. $2.37307$
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  + (−0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.258 + 0.965i)5-s + i·6-s + 8-s + (0.5 + 0.866i)9-s + (0.965 − 0.258i)10-s + (0.258 + 0.965i)11-s + (0.866 − 0.5i)12-s + (0.707 − 0.707i)13-s + (0.707 − 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.965 + 0.258i)17-s + (0.5 − 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + (−0.258 + 0.965i)5-s + i·6-s + 8-s + (0.5 + 0.866i)9-s + (0.965 − 0.258i)10-s + (0.258 + 0.965i)11-s + (0.866 − 0.5i)12-s + (0.707 − 0.707i)13-s + (0.707 − 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.965 + 0.258i)17-s + (0.5 − 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(511\)    =    \(7 \cdot 73\)
Sign: $0.284 + 0.958i$
Analytic conductor: \(2.37307\)
Root analytic conductor: \(2.37307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{511} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 511,\ (0:\ ),\ 0.284 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3377906796 + 0.2520539456i\)
\(L(\frac12)\) \(\approx\) \(0.3377906796 + 0.2520539456i\)
\(L(1)\) \(\approx\) \(0.5327498521 - 0.06877580914i\)
\(L(1)\) \(\approx\) \(0.5327498521 - 0.06877580914i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
73 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-0.258 + 0.965i)T \)
11 \( 1 + (0.258 + 0.965i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-0.965 + 0.258i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 + (0.258 + 0.965i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 + (0.965 + 0.258i)T \)
53 \( 1 + (0.965 - 0.258i)T \)
59 \( 1 + (-0.965 + 0.258i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 - T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + iT \)
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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

−23.6590672909297693077206741728, −22.70434510406190370544651124145, −21.98501038553088805283667401777, −20.84992344524668724106571627611, −20.07701183136576161299710627423, −18.861904126041291861332210185943, −18.22432228393770132292274744392, −17.10076123320607126360347828058, −16.6375498849103818478705487356, −15.92906654376240361841889367131, −15.37277097191862635380371052698, −13.98984359364649376149309765615, −13.2759318989807983438343512779, −11.88737984589643573839233247319, −11.25859231474318391391341666796, −10.13654768166946010845139336224, −9.17662242017751961328518265441, −8.598285750689105012666695134778, −7.44235377339796167978657483736, −6.19128016852160302041848852872, −5.723340874793128326725705088185, −4.55861983993543585377673687793, −3.90745644695973723135038891860, −1.50207914064707660821018140695, −0.338796268832582986951017010213, 1.3492855875507755940697494073, 2.38783187710481671708455577702, 3.584486096315136342171028558066, 4.65578053315093558678566648464, 5.979234055071680556072661312914, 7.14808507257232904225697745685, 7.63044969540094558913524651895, 8.96102657834009860028113041504, 10.160251126594738630624092080349, 10.77101881427815057017138481141, 11.50851991059883630616679416614, 12.27966219207481078902452111842, 13.16135461770178783627068142523, 14.02801483113091052499255180624, 15.416906511561670312610351659604, 16.2224318997763167776155782492, 17.57361988974266837823886379104, 17.891102627280627751510777036361, 18.45910881268255672154672330597, 19.637103378619095911314788983148, 20.02228762893744526043515067190, 21.38547133829990235012472092772, 22.22352237207164096312016872123, 22.70983563126349695435066609267, 23.39186131635365105416694054490

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