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Properties

Label 1-511-511.130-r0-0-0
Degree $1$
Conductor $511$
Sign $-0.204 - 0.978i$
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  + (0.766 − 0.642i)2-s + (−0.5 + 0.866i)3-s + (0.173 − 0.984i)4-s + (−0.173 + 0.984i)5-s + (0.173 + 0.984i)6-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.766 − 0.642i)11-s + (0.766 + 0.642i)12-s + (−0.173 − 0.984i)13-s + (−0.766 − 0.642i)15-s + (−0.939 − 0.342i)16-s − 17-s + (−0.939 − 0.342i)18-s + (0.766 − 0.642i)19-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)2-s + (−0.5 + 0.866i)3-s + (0.173 − 0.984i)4-s + (−0.173 + 0.984i)5-s + (0.173 + 0.984i)6-s + (−0.5 − 0.866i)8-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)10-s + (−0.766 − 0.642i)11-s + (0.766 + 0.642i)12-s + (−0.173 − 0.984i)13-s + (−0.766 − 0.642i)15-s + (−0.939 − 0.342i)16-s − 17-s + (−0.939 − 0.342i)18-s + (0.766 − 0.642i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(511\)    =    \(7 \cdot 73\)
Sign: $-0.204 - 0.978i$
Analytic conductor: \(2.37307\)
Root analytic conductor: \(2.37307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{511} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 511,\ (0:\ ),\ -0.204 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7149526318 - 0.8793025009i\)
\(L(\frac12)\) \(\approx\) \(0.7149526318 - 0.8793025009i\)
\(L(1)\) \(\approx\) \(1.048994628 - 0.3177950105i\)
\(L(1)\) \(\approx\) \(1.048994628 - 0.3177950105i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
73 \( 1 \)
good2 \( 1 + (0.766 - 0.642i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (-0.766 - 0.642i)T \)
13 \( 1 + (-0.173 - 0.984i)T \)
17 \( 1 - T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (0.939 + 0.342i)T \)
31 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−23.69176616922665515151319539184, −23.36528678080881342412985581614, −22.38648969631929070900743590316, −21.44954878821280382432138319065, −20.56893427945294689720189386550, −19.73459059457528481977901748494, −18.59230097134598518693092252157, −17.54404927842268048021347133395, −17.0903137909577628260821146909, −16.05549170049330345688469351578, −15.52655964172998719462240076981, −14.11204998861701474366624635107, −13.42584396373666142816319328578, −12.73981028865787037468566910259, −11.92713265197061274325624761992, −11.3564816366331435575981087487, −9.68626564001936426879020632524, −8.44384325287279540412886204026, −7.74392578477398376797384594548, −6.87157323688957975677242268528, −5.91435200259111426166659402745, −4.95727728988941481435205487070, −4.337861998978152345636342184796, −2.68921055291878260673604587159, −1.52760755891202200362184531620, 0.494429156197596307636202377101, 2.65311892894996096094944821866, 3.11735088085748742064645345760, 4.29798075251375898472197112521, 5.20316168911602426256680138879, 6.082602168521578724588298631087, 6.99866396541023815626444797417, 8.54198846606301230187308731299, 9.812000993362321582425229857168, 10.599186342844720551745566541019, 10.998461576573776162504928328936, 11.89218672405209946263057256227, 12.95388000564156388193683182828, 13.96549576590088013239671826823, 14.783048592649676378645481826583, 15.63359711000530738801513319738, 16.0306370752357188233842021307, 17.71331951625570825594598704490, 18.18659117854867576762500519303, 19.39103988602839230000467433571, 20.12055955029091572678453607022, 21.10566108790858509410989288483, 21.72969869507065295560560690090, 22.58579212420414793352262611154, 22.8581885588157635353735254875

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