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Properties

Label 1-287-287.115-r1-0-0
Degree $1$
Conductor $287$
Sign $0.698 - 0.715i$
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 + (0.866 − 0.5i)3-s + (0.913 + 0.406i)4-s + (−0.104 − 0.994i)5-s + (0.951 − 0.309i)6-s + (0.809 + 0.587i)8-s + (0.5 − 0.866i)9-s + (0.104 − 0.994i)10-s + (0.994 + 0.104i)11-s + (0.994 − 0.104i)12-s + (−0.951 + 0.309i)13-s + (−0.587 − 0.809i)15-s + (0.669 + 0.743i)16-s + (0.994 + 0.104i)17-s + (0.669 − 0.743i)18-s + (0.743 − 0.669i)19-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (0.866 − 0.5i)3-s + (0.913 + 0.406i)4-s + (−0.104 − 0.994i)5-s + (0.951 − 0.309i)6-s + (0.809 + 0.587i)8-s + (0.5 − 0.866i)9-s + (0.104 − 0.994i)10-s + (0.994 + 0.104i)11-s + (0.994 − 0.104i)12-s + (−0.951 + 0.309i)13-s + (−0.587 − 0.809i)15-s + (0.669 + 0.743i)16-s + (0.994 + 0.104i)17-s + (0.669 − 0.743i)18-s + (0.743 − 0.669i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.698 - 0.715i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.698 - 0.715i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.935212952 - 2.079208165i\)
\(L(\frac12)\) \(\approx\) \(4.935212952 - 2.079208165i\)
\(L(1)\) \(\approx\) \(2.648897824 - 0.5599656253i\)
\(L(1)\) \(\approx\) \(2.648897824 - 0.5599656253i\)

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 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.104 - 0.994i)T \)
11 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (0.994 + 0.104i)T \)
19 \( 1 + (0.743 - 0.669i)T \)
23 \( 1 + (-0.978 - 0.207i)T \)
29 \( 1 + (0.587 + 0.809i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (-0.207 + 0.978i)T \)
53 \( 1 + (0.406 - 0.913i)T \)
59 \( 1 + (-0.669 + 0.743i)T \)
61 \( 1 + (0.669 + 0.743i)T \)
67 \( 1 + (-0.406 + 0.913i)T \)
71 \( 1 + (-0.587 + 0.809i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.743 + 0.669i)T \)
97 \( 1 + (0.587 + 0.809i)T \)
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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

−25.142742302102702340607903523947, −24.77004065942382405338562756834, −23.40575211799657557234098412551, −22.442589653083892632754820290472, −21.91572681317346251455161643590, −21.08196609651673515363907426780, −19.941216671271205809237714174792, −19.48181307459496625183024352181, −18.47349680020395523690079777858, −16.88094834673885379162636386485, −15.830544823287444704173151921918, −14.96991504780708941849746410296, −14.24604107271386343842997486995, −13.83231003955045248804695443647, −12.30601287275025403838165245367, −11.539842399134838035426317019442, −10.177613092067669760382115202661, −9.8219699908358290898157897695, −8.010963670730919538503979625766, −7.16079600316368264729211882976, −5.98539409163847595080612117489, −4.690001352062306614371566582, −3.555452495429250778963097922809, −2.957068064109524813799069823304, −1.69101749761411737815796810574, 1.15801501838789296556381221880, 2.315273545616611076075796106874, 3.63020582175138484585754538148, 4.51775111979513598518395167943, 5.73060537182568020539314838051, 6.99683637355254964992977055973, 7.787305548458084929738572608548, 8.87810402494040941162921332700, 9.88955403320149764683001233928, 11.820321136786389473029743754737, 12.20620688280059344079433825312, 13.16692162262420084842143883795, 14.121113973276725329083568972318, 14.70223067186395193101318353929, 15.85673851632784542765057949521, 16.72286597963951940597103739558, 17.70647429471926676961903523562, 19.3204276712634002743856740626, 19.88583115986198287621455833360, 20.62850678203906272009334559655, 21.539499589781629945778698149, 22.47248454863515192664096781085, 23.79356845575862986881743072445, 24.21701152947729715407664588558, 24.97718543850032369865030590079

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