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Properties

Label 1-287-287.277-r0-0-0
Degree $1$
Conductor $287$
Sign $0.211 - 0.977i$
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  + (−0.978 − 0.207i)2-s + (0.5 + 0.866i)3-s + (0.913 + 0.406i)4-s + (−0.104 − 0.994i)5-s + (−0.309 − 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.5 + 0.866i)9-s + (−0.104 + 0.994i)10-s + (0.104 − 0.994i)11-s + (0.104 + 0.994i)12-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)15-s + (0.669 + 0.743i)16-s + (0.104 − 0.994i)17-s + (0.669 − 0.743i)18-s + (−0.669 − 0.743i)19-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.5 + 0.866i)3-s + (0.913 + 0.406i)4-s + (−0.104 − 0.994i)5-s + (−0.309 − 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.5 + 0.866i)9-s + (−0.104 + 0.994i)10-s + (0.104 − 0.994i)11-s + (0.104 + 0.994i)12-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)15-s + (0.669 + 0.743i)16-s + (0.104 − 0.994i)17-s + (0.669 − 0.743i)18-s + (−0.669 − 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.211 - 0.977i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.211 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5636663537 - 0.4548390947i\)
\(L(\frac12)\) \(\approx\) \(0.5636663537 - 0.4548390947i\)
\(L(1)\) \(\approx\) \(0.7151456948 - 0.1413970928i\)
\(L(1)\) \(\approx\) \(0.7151456948 - 0.1413970928i\)

Euler product

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

−25.74891351296343058027809635085, −25.224497131083368525511220790726, −23.8919792728944777389549433078, −23.51834269588917648537186205592, −22.13204005135672534927248281989, −20.88438040036701830075872444711, −19.90327863973465534269963980974, −19.13667738565622025981117317059, −18.570721054021355117617103765190, −17.67140345059952459831524608541, −16.921793900021716629272395444826, −15.44049367051872776603511266024, −14.72408041986795496680988931125, −14.00830241743465754448566275826, −12.42318561432120776045017962528, −11.7142824578052142115673074610, −10.46167063722041230217314925941, −9.628960423541878397937530308291, −8.435491040521603427469508056783, −7.571229351808125171793971484857, −6.76562389422130278629018223727, −6.04835623886194117774867841858, −3.86561740249184078141677385983, −2.39479337740722638086995432640, −1.71400594747698156402009482698, 0.60158371882890165811469956946, 2.37513287563077915070298326082, 3.44586499695532227623401703786, 4.756346885796283926466276453555, 5.95473518103661561300321485018, 7.65060601935214442552182904103, 8.468436618336836712619377102233, 9.12631291081263348441438338621, 10.07435693985155484325612251392, 11.00096695829080886992099752849, 12.023992745778788358628272264332, 13.174094894367098752869125593096, 14.37358719098210106021060139268, 15.796071031244447264248962107265, 16.00628411001206765851815064209, 17.04215177745661510300171040185, 17.89791580092606560144427481868, 19.35809245746059357763545945207, 19.79885912061652966649476891880, 20.71237776185804602859426346860, 21.34497200080387456964681005513, 22.28512977131830372842753674112, 23.81848537283674472063098616867, 24.89186334258681917293451448686, 25.28822634624453770923439869368

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