Properties

Label 1-287-287.151-r1-0-0
Degree $1$
Conductor $287$
Sign $0.860 + 0.508i$
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  + (−0.994 + 0.104i)2-s + (0.258 − 0.965i)3-s + (0.978 − 0.207i)4-s + (−0.743 + 0.669i)5-s + (−0.156 + 0.987i)6-s + (−0.951 + 0.309i)8-s + (−0.866 − 0.5i)9-s + (0.669 − 0.743i)10-s + (−0.998 + 0.0523i)11-s + (0.0523 − 0.998i)12-s + (−0.987 − 0.156i)13-s + (0.453 + 0.891i)15-s + (0.913 − 0.406i)16-s + (−0.0523 − 0.998i)17-s + (0.913 + 0.406i)18-s + (0.358 − 0.933i)19-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)2-s + (0.258 − 0.965i)3-s + (0.978 − 0.207i)4-s + (−0.743 + 0.669i)5-s + (−0.156 + 0.987i)6-s + (−0.951 + 0.309i)8-s + (−0.866 − 0.5i)9-s + (0.669 − 0.743i)10-s + (−0.998 + 0.0523i)11-s + (0.0523 − 0.998i)12-s + (−0.987 − 0.156i)13-s + (0.453 + 0.891i)15-s + (0.913 − 0.406i)16-s + (−0.0523 − 0.998i)17-s + (0.913 + 0.406i)18-s + (0.358 − 0.933i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.860 + 0.508i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.860 + 0.508i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5565251024 + 0.1521280438i\)
\(L(\frac12)\) \(\approx\) \(0.5565251024 + 0.1521280438i\)
\(L(1)\) \(\approx\) \(0.5483122681 - 0.08560384696i\)
\(L(1)\) \(\approx\) \(0.5483122681 - 0.08560384696i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.994 - 0.104i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (0.743 - 0.669i)T \)
11 \( 1 + (0.998 - 0.0523i)T \)
13 \( 1 + (0.987 + 0.156i)T \)
17 \( 1 + (0.0523 + 0.998i)T \)
19 \( 1 + (-0.358 + 0.933i)T \)
23 \( 1 + (-0.104 - 0.994i)T \)
29 \( 1 + (-0.891 + 0.453i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (-0.669 - 0.743i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (0.777 + 0.629i)T \)
53 \( 1 + (-0.838 - 0.544i)T \)
59 \( 1 + (-0.913 - 0.406i)T \)
61 \( 1 + (0.406 + 0.913i)T \)
67 \( 1 + (0.544 - 0.838i)T \)
71 \( 1 + (0.453 - 0.891i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.965 + 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.933 - 0.358i)T \)
97 \( 1 + (-0.453 - 0.891i)T \)
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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

−25.489268561023331448125423131569, −24.47783571258197906907906493523, −23.63589266173605845677475783349, −22.331084145504870415055418802053, −21.20090431065256089657511594935, −20.641473823720459130986700568204, −19.756062993268722315903449139728, −19.1071871534047506652213479239, −17.8604122124178190706729758168, −16.64289674215412498084273754944, −16.3440078278977212598548908676, −15.30562358457634639775059626136, −14.59735894312980313222436008845, −12.81492611327549897272201347413, −11.94462085181382238814112714348, −10.80208230452565539576928084412, −10.11735249091048055130786429437, −9.06483214490659083800244575538, −8.22213660742032092200286144419, −7.51454730656391948357801875301, −5.799385652116667542232573641055, −4.60315056805671656177859074971, −3.42261839863024524434719929785, −2.189005989714616765612569316181, −0.33989365070027008758536017607, 0.7512317175543986577463902468, 2.452316589645267983818765340737, 3.0279019529313712401670385717, 5.202811158998985505700818346, 6.64277952333658535494619235773, 7.41591109804967835698306182614, 7.907092527154421765925431501589, 9.12586769381531360195704475396, 10.24863942863653149274291691503, 11.40989635370373893594623747933, 11.98025193405881836677237603672, 13.24754956034363923425172504751, 14.47981856047388142698822906261, 15.3438207898978286551855384790, 16.20014547626919685911646151586, 17.6167471712377307184897930757, 18.078742180139008356315892073321, 18.98232273358924663534262665202, 19.687913497860877119327016287138, 20.32345843208402740117015869411, 21.67575435667829170494941429060, 23.06332919894118444230646518905, 23.74834519654061569093108163251, 24.59964161752117316326497383321, 25.48018088945630521297117810222

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