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Properties

Label 1-287-287.221-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.964 + 0.263i$
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  + (0.913 − 0.406i)2-s + (−0.5 − 0.866i)3-s + (0.669 − 0.743i)4-s + (−0.978 − 0.207i)5-s + (−0.809 − 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)9-s + (−0.978 + 0.207i)10-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)15-s + (−0.104 − 0.994i)16-s + (−0.978 + 0.207i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (−0.5 − 0.866i)3-s + (0.669 − 0.743i)4-s + (−0.978 − 0.207i)5-s + (−0.809 − 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)9-s + (−0.978 + 0.207i)10-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)15-s + (−0.104 − 0.994i)16-s + (−0.978 + 0.207i)17-s + (−0.104 + 0.994i)18-s + (−0.104 − 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.964 + 0.263i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.964 + 0.263i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1144656053 - 0.8533545264i\)
\(L(\frac12)\) \(\approx\) \(-0.1144656053 - 0.8533545264i\)
\(L(1)\) \(\approx\) \(0.7296991213 - 0.6941626438i\)
\(L(1)\) \(\approx\) \(0.7296991213 - 0.6941626438i\)

Euler product

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

−26.32264728843180743309974865942, −24.90749930434829470432747048370, −23.87109759329443775697469488069, −23.270864361046494712677629345871, −22.51659586872659677165435718977, −21.69003779597469243252270760545, −20.853951280955114420210710576574, −20.011486110198300842186573230996, −18.78060931859867392197561658341, −17.408722974129690261400600452823, −16.53265500064497044817674323159, −15.74010639710359692634148150815, −15.155530158967587187191501177257, −14.26993122799075570856182382021, −12.95748483311713415933385872464, −11.93327155582593157884462032580, −11.26290113465069844905322646127, −10.34017763379906754212696932738, −8.822928818893305814523426107535, −7.666567354936176851481797674505, −6.695218863913368370037586655161, −5.45138977444894266147049224764, −4.587094839982940701800756325404, −3.73772415171416504504131979157, −2.61672558327888141271382532914, 0.4204095712598869664469704815, 2.10941864735201500330799830867, 3.165826882032406659727267674393, 4.75522246695909033744241850962, 5.28131020461786617618098996596, 6.841143272919970854755969664827, 7.375897634884406351473888933234, 8.70566583143690834431532328809, 10.56528999010605084425421665290, 11.096008978452299627468643656472, 12.22912750848739219144430582310, 12.756529155991835835544625994583, 13.54182398942861450435259281208, 14.91077967839310184284572225235, 15.60163972738413640897310166974, 16.66684631734971289282269942331, 17.898410612663674276532074499187, 18.882503798394406267896553785965, 19.775162179048146289575526513919, 20.26928107829693191877915434753, 21.67007305451524538629140692160, 22.53119144222849251604217313917, 23.27435935253706566867305532872, 24.0071688106129569906726500534, 24.51170448803275681928744717729

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