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Properties

Label 1-287-287.250-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.799 - 0.600i$
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  + (−0.978 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.799 - 0.600i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (250, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.799 - 0.600i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06773620401 - 0.2029789924i\)
\(L(\frac12)\) \(\approx\) \(0.06773620401 - 0.2029789924i\)
\(L(1)\) \(\approx\) \(0.5248297695 + 0.03441218601i\)
\(L(1)\) \(\approx\) \(0.5248297695 + 0.03441218601i\)

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

−25.94427699118031026461904777866, −24.735530312367725846263738816631, −24.11885883107863034210614488260, −23.01667713500149404663318017103, −21.93896224862305617946948477526, −21.22199588152934688352518045281, −19.71750960758030084401248237938, −19.06694498442919491706668581179, −18.44783643385573496842018347496, −17.69364119818226977518908840970, −16.719828952370383680175319772018, −15.9340483853256605406459852445, −14.48010829358209696852202669317, −13.5987985683459847732308253956, −12.26293354147655004311582496892, −11.41154004384340333948056149329, −10.798186732155262843200785175, −9.71717273913025110115584350584, −8.32584293554425008730328735261, −7.62026929495260118132950619209, −6.34632758553655442717935315595, −6.07950763281785000098986581216, −3.713838675521860281474568555924, −2.42890451170863578849170811389, −1.406830831040168409871001643891, 0.10347007537151730662230254701, 1.30342262135977416755482752642, 3.02390587193569231948021753571, 4.70789666659979471570470566509, 5.43128714811480358503259350947, 6.66756802805514889979237611003, 7.94319170505162505502926391439, 9.01495477220220732236114655091, 9.7078910620771669801479555209, 10.52105588604610001419145388860, 11.714764885547263031572890729886, 12.4098225379166099550020423751, 14.00348369668774070270674910658, 15.42631922089207623782227413355, 15.80878247137412432171438910312, 16.697569436217990997857778641180, 17.72463562780237433264522065039, 17.99475588350815133455481402325, 19.86153510539550141723975081377, 20.26557426491896951534162913705, 21.01586322092865395707507116618, 22.204740773308348608921748730188, 23.26396259338193107113706505668, 24.15445329474793448064341586622, 25.22828768208809949063224218186

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