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Properties

Label 1-511-511.131-r0-0-0
Degree $1$
Conductor $511$
Sign $0.958 + 0.286i$
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 + (0.0871 − 0.996i)5-s + (0.642 − 0.766i)6-s + (−0.5 + 0.866i)8-s + (0.5 − 0.866i)9-s + (0.258 + 0.965i)10-s + (0.422 + 0.906i)11-s + (−0.342 + 0.939i)12-s + (−0.0871 − 0.996i)13-s + (0.422 + 0.906i)15-s + (0.173 − 0.984i)16-s + (−0.707 + 0.707i)17-s + (−0.173 + 0.984i)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 + (0.0871 − 0.996i)5-s + (0.642 − 0.766i)6-s + (−0.5 + 0.866i)8-s + (0.5 − 0.866i)9-s + (0.258 + 0.965i)10-s + (0.422 + 0.906i)11-s + (−0.342 + 0.939i)12-s + (−0.0871 − 0.996i)13-s + (0.422 + 0.906i)15-s + (0.173 − 0.984i)16-s + (−0.707 + 0.707i)17-s + (−0.173 + 0.984i)18-s + (0.342 + 0.939i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(511\)    =    \(7 \cdot 73\)
Sign: $0.958 + 0.286i$
Analytic conductor: \(2.37307\)
Root analytic conductor: \(2.37307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{511} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 511,\ (0:\ ),\ 0.958 + 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6335073162 + 0.09266761651i\)
\(L(\frac12)\) \(\approx\) \(0.6335073162 + 0.09266761651i\)
\(L(1)\) \(\approx\) \(0.5805556081 + 0.07363447412i\)
\(L(1)\) \(\approx\) \(0.5805556081 + 0.07363447412i\)

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 \( 1 + (0.0871 - 0.996i)T \)
11 \( 1 + (0.422 + 0.906i)T \)
13 \( 1 + (-0.0871 - 0.996i)T \)
17 \( 1 + (-0.707 + 0.707i)T \)
19 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (0.819 - 0.573i)T \)
31 \( 1 + (-0.819 + 0.573i)T \)
37 \( 1 + (0.766 - 0.642i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (-0.965 + 0.258i)T \)
47 \( 1 + (-0.422 - 0.906i)T \)
53 \( 1 + (0.996 - 0.0871i)T \)
59 \( 1 + (0.422 - 0.906i)T \)
61 \( 1 + (0.342 - 0.939i)T \)
67 \( 1 + (0.342 + 0.939i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
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

−23.69493661332465971107752138863, −22.41542396308662456643899299102, −21.9602983104850575166964484886, −21.167787313875411197757682475777, −19.7348474492174194933476716268, −19.19962061061643556651843575183, −18.24470331619097478048462980940, −18.01425723440784669011800979517, −16.78028119754071096901769883566, −16.34576514272639855438041757856, −15.20923959395069890923017380485, −13.9543282436103041337136866037, −13.07737543882228597300184268344, −11.779854434572343857343050457536, −11.32081814103864669462342077269, −10.72600093151453577667039968817, −9.610092493827964325261014073964, −8.67097387805055239365956100243, −7.38073731281139189161584874200, −6.76586998755144463905288103550, −6.12154501638360263887549602307, −4.540773744815200651889578226968, −3.06596152192767357565709503297, −2.14221502965830561578960212499, −0.80812674480107745408954239011, 0.81746154251537822349929454053, 1.89589389586802582527372944013, 3.79607554674193408070259114295, 5.022487847795773834433716186388, 5.68394249211793245801701199581, 6.70541793602649355793604442507, 7.78790004089752116255550514946, 8.80309951359870011925844685328, 9.68373550607269871553129583172, 10.28514344649518368219085253309, 11.33837979413758986055429246472, 12.21523867987108001319782219629, 12.98611675524940485509546492688, 14.63947261320638741438715779251, 15.4295114460704829793201063464, 16.11938793520623595630717617451, 16.97678941637288287929545362806, 17.556934921541212258466534277614, 18.10386264939293410869121860285, 19.517909113701876735145354858871, 20.19223911755370595617934442515, 20.93277110274518467974925962724, 21.87296708087410646252014048528, 23.12138814479909936040366791849, 23.52552913580767284731645970621

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