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Properties

Label 1-287-287.244-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.542 + 0.839i$
Analytic cond. $30.8424$
Root an. cond. $30.8424$
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  + (0.809 − 0.587i)2-s − i·3-s + (0.309 − 0.951i)4-s + (0.309 − 0.951i)5-s + (−0.587 − 0.809i)6-s + (−0.309 − 0.951i)8-s − 9-s + (−0.309 − 0.951i)10-s + (−0.951 + 0.309i)11-s + (−0.951 − 0.309i)12-s + (0.587 + 0.809i)13-s + (−0.951 − 0.309i)15-s + (−0.809 − 0.587i)16-s + (−0.951 + 0.309i)17-s + (−0.809 + 0.587i)18-s + (0.587 − 0.809i)19-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s − i·3-s + (0.309 − 0.951i)4-s + (0.309 − 0.951i)5-s + (−0.587 − 0.809i)6-s + (−0.309 − 0.951i)8-s − 9-s + (−0.309 − 0.951i)10-s + (−0.951 + 0.309i)11-s + (−0.951 − 0.309i)12-s + (0.587 + 0.809i)13-s + (−0.951 − 0.309i)15-s + (−0.809 − 0.587i)16-s + (−0.951 + 0.309i)17-s + (−0.809 + 0.587i)18-s + (0.587 − 0.809i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.542 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.542 + 0.839i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.542 + 0.839i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9350618331 - 1.717372637i\)
\(L(\frac12)\) \(\approx\) \(-0.9350618331 - 1.717372637i\)
\(L(1)\) \(\approx\) \(0.7717798044 - 1.200422678i\)
\(L(1)\) \(\approx\) \(0.7717798044 - 1.200422678i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 - iT \)
5 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-0.951 + 0.309i)T \)
13 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (0.587 - 0.809i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (0.951 + 0.309i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + (-0.951 - 0.309i)T \)
71 \( 1 + (-0.951 + 0.309i)T \)
73 \( 1 + T \)
79 \( 1 + iT \)
83 \( 1 - T \)
89 \( 1 + (-0.587 + 0.809i)T \)
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

−25.85234688799019884749440943257, −25.19175823922031937967664178431, −23.96339632542312413649770929139, −22.77334322361940895204711411652, −22.55761708743366125047511131150, −21.49462823105954024839597332969, −20.87103589446352260990157442062, −19.912174702777444595835476611449, −18.22188247350123633209660228982, −17.655234935000399179933762908727, −16.24175688650929855476520229979, −15.76984626938922277574164000391, −14.87684749422680603109030963312, −14.041189620002678805257692471743, −13.25658834529220630305284138270, −11.835546541944721791052277535868, −10.82445728543810623467521032575, −10.14076357007636717560047476186, −8.63023582393047252886959094434, −7.70759447343141624661360426638, −6.3134868090658279561298637723, −5.611308271944046681593345252278, −4.487627428733055279703774912717, −3.29191054181739548610322182389, −2.61180667968559229032392791677, 0.41227602385267540015099758090, 1.68058640817009150586869404097, 2.49418055108941569634717033691, 4.10372210809343905007874482680, 5.23288104286312582568895679260, 6.097346143479331241958172419539, 7.22169341565455829282680750328, 8.55675995839385244205189864035, 9.55816660647735516833035152538, 10.94645289913950038628695143003, 11.83268677043688069888600250635, 12.72058231699594210286494394444, 13.43289313746787716070257800870, 13.95652395298514453285297655143, 15.405521897674435621285296656797, 16.29948593285286741468377962532, 17.68768404098010678752401102497, 18.363206972047111865077016845046, 19.5786340957894615853680207587, 20.133039763243079511881204392477, 21.0586684758378470314138118343, 21.926572866065170798457346364042, 23.11682032856320935670580593740, 23.94266774160452594695044221033, 24.24282646562050845676371245289

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