Properties

Label 1-287-287.222-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.987 + 0.155i$
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  + (0.743 + 0.669i)2-s + (0.258 + 0.965i)3-s + (0.104 + 0.994i)4-s + (−0.406 + 0.913i)5-s + (−0.453 + 0.891i)6-s + (−0.587 + 0.809i)8-s + (−0.866 + 0.5i)9-s + (−0.913 + 0.406i)10-s + (0.358 − 0.933i)11-s + (−0.933 + 0.358i)12-s + (0.891 + 0.453i)13-s + (−0.987 − 0.156i)15-s + (−0.978 + 0.207i)16-s + (0.933 + 0.358i)17-s + (−0.978 − 0.207i)18-s + (−0.838 − 0.544i)19-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (0.258 + 0.965i)3-s + (0.104 + 0.994i)4-s + (−0.406 + 0.913i)5-s + (−0.453 + 0.891i)6-s + (−0.587 + 0.809i)8-s + (−0.866 + 0.5i)9-s + (−0.913 + 0.406i)10-s + (0.358 − 0.933i)11-s + (−0.933 + 0.358i)12-s + (0.891 + 0.453i)13-s + (−0.987 − 0.156i)15-s + (−0.978 + 0.207i)16-s + (0.933 + 0.358i)17-s + (−0.978 − 0.207i)18-s + (−0.838 − 0.544i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.987 + 0.155i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.987 + 0.155i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1383269282 + 1.767047698i\)
\(L(\frac12)\) \(\approx\) \(0.1383269282 + 1.767047698i\)
\(L(1)\) \(\approx\) \(0.8873344210 + 1.224665785i\)
\(L(1)\) \(\approx\) \(0.8873344210 + 1.224665785i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.743 + 0.669i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-0.406 + 0.913i)T \)
11 \( 1 + (0.358 - 0.933i)T \)
13 \( 1 + (0.891 + 0.453i)T \)
17 \( 1 + (0.933 + 0.358i)T \)
19 \( 1 + (-0.838 - 0.544i)T \)
23 \( 1 + (-0.669 + 0.743i)T \)
29 \( 1 + (0.156 - 0.987i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (0.998 - 0.0523i)T \)
53 \( 1 + (-0.777 - 0.629i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (0.207 + 0.978i)T \)
67 \( 1 + (-0.629 + 0.777i)T \)
71 \( 1 + (0.987 - 0.156i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.965 - 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.544 + 0.838i)T \)
97 \( 1 + (0.987 + 0.156i)T \)
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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

−25.0907853110244079219764784543, −23.93659118966301044851488890379, −23.319232647862220745905528330552, −22.720534230333756879553915627141, −21.23796146574945990099530296319, −20.36719896973107825109333046364, −19.97161163156047151352749528508, −18.9057785434050881000979593434, −18.12825493511777118068327710420, −16.88153440793871611636168023094, −15.6822598438856929799250497930, −14.623876523954022852070122905576, −13.802856019507715766693072919920, −12.55378630025816556642245352780, −12.51271946149925689548307973037, −11.4157432300232138446756134744, −10.11068269397978082224768756228, −8.90233966654758170737957955689, −7.93375927556777781736235960876, −6.63375706759186172413866352634, −5.61758707762939558430092519322, −4.41568100820911920300607664492, −3.33104128180679101918802455032, −1.93534420239914325153486223536, −0.9588939610717503364154962505, 2.63882313762772328123513347051, 3.63703515175551715606787855814, 4.237790385953707225330637521332, 5.75703578231088771506681150160, 6.45780457979370487768473059704, 7.909237961524668868532148580401, 8.597212151227597982387778205466, 9.974131711352799741622515442800, 11.21481228673107856018406297953, 11.702789555745540370936029451724, 13.42769004934780748492705174024, 14.11668017035406032378791116126, 14.95656527182012268214728764286, 15.67744108200730333109405012354, 16.47634740476771937046131834135, 17.38025309252922236408189253540, 18.757076688352700999643513774228, 19.67574301133615614781661583525, 21.013026079457929268190670012862, 21.56385657637527733681953686528, 22.32205732538594075833436647617, 23.232101493509230195874326621867, 23.899576737921807145256541754063, 25.32842625208521415970052269223, 25.88737664995662663334007995721

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