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Properties

Label 1-511-511.160-r0-0-0
Degree $1$
Conductor $511$
Sign $0.961 - 0.275i$
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  + (−0.939 − 0.342i)2-s + (0.866 − 0.5i)3-s + (0.766 + 0.642i)4-s + (0.906 + 0.422i)5-s + (−0.984 + 0.173i)6-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (−0.707 − 0.707i)10-s + (−0.422 + 0.906i)11-s + (0.984 + 0.173i)12-s + (0.819 + 0.573i)13-s + (0.996 − 0.0871i)15-s + (0.173 + 0.984i)16-s + (−0.965 + 0.258i)17-s + (−0.766 + 0.642i)18-s + (0.342 − 0.939i)19-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.866 − 0.5i)3-s + (0.766 + 0.642i)4-s + (0.906 + 0.422i)5-s + (−0.984 + 0.173i)6-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (−0.707 − 0.707i)10-s + (−0.422 + 0.906i)11-s + (0.984 + 0.173i)12-s + (0.819 + 0.573i)13-s + (0.996 − 0.0871i)15-s + (0.173 + 0.984i)16-s + (−0.965 + 0.258i)17-s + (−0.766 + 0.642i)18-s + (0.342 − 0.939i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(511\)    =    \(7 \cdot 73\)
Sign: $0.961 - 0.275i$
Analytic conductor: \(2.37307\)
Root analytic conductor: \(2.37307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{511} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 511,\ (0:\ ),\ 0.961 - 0.275i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.451187331 - 0.2035657538i\)
\(L(\frac12)\) \(\approx\) \(1.451187331 - 0.2035657538i\)
\(L(1)\) \(\approx\) \(1.113592079 - 0.1676330250i\)
\(L(1)\) \(\approx\) \(1.113592079 - 0.1676330250i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
73 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (0.906 + 0.422i)T \)
11 \( 1 + (-0.422 + 0.906i)T \)
13 \( 1 + (0.819 + 0.573i)T \)
17 \( 1 + (-0.965 + 0.258i)T \)
19 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (0.342 + 0.939i)T \)
29 \( 1 + (0.906 - 0.422i)T \)
31 \( 1 + (0.0871 - 0.996i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.965 + 0.258i)T \)
47 \( 1 + (0.573 + 0.819i)T \)
53 \( 1 + (0.422 + 0.906i)T \)
59 \( 1 + (-0.573 + 0.819i)T \)
61 \( 1 + (-0.984 - 0.173i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (0.984 - 0.173i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−24.093390349163268305124527380990, −22.784842105926560255068700901656, −21.55638976917552865643931243759, −20.89295344156621592873781294094, −20.29865505668135349504839408294, −19.41752303108984498420549248718, −18.41290223422744580002944954897, −17.863094641320624719610902863640, −16.601185226701833568906628338256, −16.12453561127316073749805037616, −15.30851530794483808602099173378, −14.15576116966152392631446359038, −13.66305060823450220128767760614, −12.4848860653942390768271481907, −10.81866355695508550305934065433, −10.46433437524737640071274432642, −9.40993410274362781307650075890, −8.60231714378184100886464864317, −8.21353576459567711250426391392, −6.81749461237175437887624868453, −5.79715981192567769605006799713, −4.881950419036261943532230987453, −3.25805762589946240352123907397, −2.30318925370240456303256711233, −1.137732708451503817197656576235, 1.32801013793877052972396628694, 2.20658892829447069456834481885, 2.92115443616322205538948585506, 4.24443776057236411188328758927, 6.07705490985204564389370145276, 6.941686767717728255516311847749, 7.61518579891292482671366314394, 8.90447277963561983428175052659, 9.29250244633527473672659563614, 10.28416852785520684798304587452, 11.18824677926061078070946191799, 12.32523129718220733517516246605, 13.31154970680759028973058481156, 13.83899922688387785346512594581, 15.2506310533175504762390019595, 15.67568139298753781644968858452, 17.28068032458241016330568690310, 17.73656567020631589177547962715, 18.48548935142025720358102686374, 19.24790216836254331300172969968, 20.09169706057711442836267110028, 20.911293157554674836428490234660, 21.42961474466161397996797525980, 22.55659004459887410931755462925, 23.891359195923672057965137166802

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