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Properties

Label 1-287-287.207-r0-0-0
Degree $1$
Conductor $287$
Sign $0.179 + 0.983i$
Analytic cond. $1.33282$
Root an. cond. $1.33282$
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  + (0.104 + 0.994i)2-s + (−0.866 + 0.5i)3-s + (−0.978 + 0.207i)4-s + (−0.669 − 0.743i)5-s + (−0.587 − 0.809i)6-s + (−0.309 − 0.951i)8-s + (0.5 − 0.866i)9-s + (0.669 − 0.743i)10-s + (−0.743 − 0.669i)11-s + (0.743 − 0.669i)12-s + (0.587 + 0.809i)13-s + (0.951 + 0.309i)15-s + (0.913 − 0.406i)16-s + (0.743 + 0.669i)17-s + (0.913 + 0.406i)18-s + (0.406 + 0.913i)19-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.866 + 0.5i)3-s + (−0.978 + 0.207i)4-s + (−0.669 − 0.743i)5-s + (−0.587 − 0.809i)6-s + (−0.309 − 0.951i)8-s + (0.5 − 0.866i)9-s + (0.669 − 0.743i)10-s + (−0.743 − 0.669i)11-s + (0.743 − 0.669i)12-s + (0.587 + 0.809i)13-s + (0.951 + 0.309i)15-s + (0.913 − 0.406i)16-s + (0.743 + 0.669i)17-s + (0.913 + 0.406i)18-s + (0.406 + 0.913i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.179 + 0.983i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.179 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5449649596 + 0.4546446294i\)
\(L(\frac12)\) \(\approx\) \(0.5449649596 + 0.4546446294i\)
\(L(1)\) \(\approx\) \(0.6151726574 + 0.3480190315i\)
\(L(1)\) \(\approx\) \(0.6151726574 + 0.3480190315i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.104 + 0.994i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.669 - 0.743i)T \)
11 \( 1 + (-0.743 - 0.669i)T \)
13 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (0.743 + 0.669i)T \)
19 \( 1 + (0.406 + 0.913i)T \)
23 \( 1 + (-0.104 - 0.994i)T \)
29 \( 1 + (-0.951 - 0.309i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (0.994 - 0.104i)T \)
53 \( 1 + (0.207 + 0.978i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (-0.207 - 0.978i)T \)
71 \( 1 + (0.951 - 0.309i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.406 - 0.913i)T \)
97 \( 1 + (0.951 + 0.309i)T \)
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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

−25.402649619903536290696434738788, −23.92463116190039369454805728905, −23.227675227032458580252180170190, −22.73044144320214507968170636596, −21.89241629999442885974274867584, −20.7911818261659191611430926145, −19.77548956296788846558402666182, −18.93626178054851069250513976682, −18.046050985169346027195986963005, −17.70835556616814228454210575008, −16.1112514993321910847780390831, −15.18096667271639842288579645729, −13.88369825684378236549852813487, −12.97124224503985499759557810080, −12.12608362544357182589555873245, −11.243231801715715023942108673228, −10.64239354405785143034142075325, −9.63287314256454813476164605193, −7.974131056528900138194367130913, −7.23365086380231315143776547375, −5.72945274236388978256710426664, −4.83735288576650678576389260974, −3.47369038176406752163319478234, −2.38246645243484250989617173607, −0.80760278721895773138599432335, 0.86738957173492508055880768808, 3.67234015863215428730378235088, 4.38784937218387349783127465834, 5.5217436530651229571288160736, 6.17950087796733698580158121548, 7.5948527687931061956907872368, 8.46239043556711076950183408547, 9.5018119141284181814189893124, 10.66168960400515102867538313741, 11.877521930995236614975978276031, 12.66896496953795132949024061421, 13.75832910833102392739658613434, 15.02878384056625535441935077755, 15.8010749232092648072315140072, 16.67637188886548336444484385135, 16.86303186824986084322804470253, 18.424101061520601528220489380294, 18.89349332474547077108528618320, 20.6843483257411938399568454648, 21.29018373873407787135446988495, 22.42629706654194777464148637857, 23.22850770584845639766558847448, 23.91848799848116487328039185436, 24.447208876761775920321217510237, 25.84850338620479433457458351624

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