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Properties

Label 1-285-285.263-r0-0-0
Degree $1$
Conductor $285$
Sign $0.188 + 0.982i$
Analytic cond. $1.32353$
Root an. cond. $1.32353$
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  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)8-s + (0.5 + 0.866i)11-s + (−0.642 + 0.766i)13-s + (−0.939 − 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.984 + 0.173i)17-s + (−0.642 − 0.766i)22-s + (0.342 + 0.939i)23-s + (0.5 − 0.866i)26-s + (0.984 + 0.173i)28-s + (0.173 − 0.984i)29-s + (−0.5 + 0.866i)31-s + (−0.642 + 0.766i)32-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)8-s + (0.5 + 0.866i)11-s + (−0.642 + 0.766i)13-s + (−0.939 − 0.342i)14-s + (0.766 − 0.642i)16-s + (−0.984 + 0.173i)17-s + (−0.642 − 0.766i)22-s + (0.342 + 0.939i)23-s + (0.5 − 0.866i)26-s + (0.984 + 0.173i)28-s + (0.173 − 0.984i)29-s + (−0.5 + 0.866i)31-s + (−0.642 + 0.766i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.188 + 0.982i$
Analytic conductor: \(1.32353\)
Root analytic conductor: \(1.32353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 285,\ (0:\ ),\ 0.188 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6061448976 + 0.5010077901i\)
\(L(\frac12)\) \(\approx\) \(0.6061448976 + 0.5010077901i\)
\(L(1)\) \(\approx\) \(0.7094834826 + 0.2189696334i\)
\(L(1)\) \(\approx\) \(0.7094834826 + 0.2189696334i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.984 + 0.173i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.642 + 0.766i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (0.342 + 0.939i)T \)
29 \( 1 + (0.173 - 0.984i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (0.984 + 0.173i)T \)
53 \( 1 + (0.342 + 0.939i)T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (-0.984 - 0.173i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.642 + 0.766i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (0.984 - 0.173i)T \)
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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

−25.42375358228428761981409646705, −24.43889708318465030843415766539, −24.047586569971066154738297771900, −22.44177657683482497330490119459, −21.60387551932825517332353306527, −20.47597192778423178711154552143, −19.99798839272123869280511527740, −18.9069389560486797817558229362, −18.0115975052685540128497175999, −17.19184747107330110563927428985, −16.50348270812623816583999193364, −15.30720541478426159101626680603, −14.42404820107189188621576909463, −13.15526183861318127381642556855, −11.94949966587817367033307537081, −11.047031332890910246773150259, −10.38963711030217157987418549246, −9.09879321707556284363063917826, −8.29539357181255287748731709997, −7.34464871068033930747116510550, −6.30638290550492413073454945625, −4.85551393181177860057317020815, −3.40076589570083699102311807972, −2.11778001537911360633830098828, −0.74430228171775677050621224701, 1.57959259880814465551021892151, 2.41313298518998615419680350953, 4.31548485845866931664026569867, 5.533964981444645621778498565152, 6.8227225109465444943979912946, 7.59218211379329072147361252951, 8.82036764707650264700675861964, 9.41935111622763114409453086435, 10.642298574557183529996178629330, 11.60998318881529782708707727232, 12.29597146905637366993929719211, 13.969233368904129840226803097201, 14.985303279697610075676494656851, 15.56020222704039748875802273436, 16.90426862903332896868561719590, 17.5352880386212311575052327922, 18.29915974024765718929179863429, 19.37851671445133930833704004892, 20.064982740964282355896272626767, 21.134068060274700653726096337085, 21.90431901426194968218471920595, 23.31439901806615378786892851639, 24.26110111893585612825265799590, 24.92314048794998578343195896870, 25.7444573192392142606742181076

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