Properties

Label 1-287-287.234-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: $\chi_{287} (234, \cdot )$
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

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

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