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Properties

Label 1-511-511.158-r0-0-0
Degree $1$
Conductor $511$
Sign $0.388 - 0.921i$
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  + (0.173 − 0.984i)2-s − 3-s + (−0.939 − 0.342i)4-s + (−0.642 − 0.766i)5-s + (−0.173 + 0.984i)6-s + (−0.5 + 0.866i)8-s + 9-s + (−0.866 + 0.5i)10-s + (−0.984 + 0.173i)11-s + (0.939 + 0.342i)12-s + (0.984 + 0.173i)13-s + (0.642 + 0.766i)15-s + (0.766 + 0.642i)16-s + (0.866 + 0.5i)17-s + (0.173 − 0.984i)18-s + (−0.173 + 0.984i)19-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s − 3-s + (−0.939 − 0.342i)4-s + (−0.642 − 0.766i)5-s + (−0.173 + 0.984i)6-s + (−0.5 + 0.866i)8-s + 9-s + (−0.866 + 0.5i)10-s + (−0.984 + 0.173i)11-s + (0.939 + 0.342i)12-s + (0.984 + 0.173i)13-s + (0.642 + 0.766i)15-s + (0.766 + 0.642i)16-s + (0.866 + 0.5i)17-s + (0.173 − 0.984i)18-s + (−0.173 + 0.984i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(511\)    =    \(7 \cdot 73\)
Sign: $0.388 - 0.921i$
Analytic conductor: \(2.37307\)
Root analytic conductor: \(2.37307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{511} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 511,\ (0:\ ),\ 0.388 - 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6030959505 - 0.4003288763i\)
\(L(\frac12)\) \(\approx\) \(0.6030959505 - 0.4003288763i\)
\(L(1)\) \(\approx\) \(0.5999761564 - 0.3492195484i\)
\(L(1)\) \(\approx\) \(0.5999761564 - 0.3492195484i\)

Euler product

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

−23.59147428409323447167995551980, −23.14508359471555981747754657745, −22.33548504356531003244220999665, −21.64817656539101840165391796249, −20.594859102636228320538917634971, −18.99383619984619123585912793211, −18.41171562631845187467933515148, −17.88182171851256954894516280552, −16.78159218221483164537150577199, −15.85507346193422263242642375972, −15.62492722384234124337593772984, −14.471188521489657061772601769209, −13.46551839720824523010995696011, −12.612844397852032659701524244403, −11.60828141568424847039985322013, −10.71268682109811393116859867102, −9.87495673947999293733131689742, −8.4201149979402042860440591114, −7.60581743578974067626946725383, −6.75383372565513095407403256, −5.96197183329689191298882197581, −5.02958688593817332386237789491, −4.05784813673567699348544223025, −2.932287259256012444381520661138, −0.660472667992060876237551934034, 0.82465982054700275159409226298, 1.872203620608756878611029628465, 3.606712402738681531704215193, 4.28352305116123172485536368289, 5.3633171238528717125964346705, 5.98867317862425138044326270375, 7.72802827954321973651374908433, 8.44954401398438653482889572685, 9.83791190232334492512613677677, 10.40992993311814129144062195570, 11.5327703410816250648437623310, 11.96268388046040227620025746233, 12.91821266783755721941217008980, 13.44920683301359408483602439192, 14.92368867895898804150409791647, 15.90642925227958941978460986400, 16.65905043196097742892118854395, 17.65048516235048714374267853621, 18.520626774770241966600974925569, 19.11375777317613544484015257006, 20.23291006153324221811427903420, 21.09491878551013117481906752756, 21.45092344427151868094586686505, 22.79924477535110589304480997729, 23.36723273332025777751308572159

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