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Properties

Label 1-287-287.109-r1-0-0
Degree $1$
Conductor $287$
Sign $0.740 - 0.672i$
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  + (−0.866 − 0.5i)2-s + (−0.965 − 0.258i)3-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s + (0.707 + 0.707i)6-s − i·8-s + (0.866 + 0.5i)9-s + (−0.5 − 0.866i)10-s + (0.965 + 0.258i)11-s + (−0.258 − 0.965i)12-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)15-s + (−0.5 + 0.866i)16-s + (0.258 − 0.965i)17-s + (−0.5 − 0.866i)18-s + (−0.965 + 0.258i)19-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.965 − 0.258i)3-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s + (0.707 + 0.707i)6-s − i·8-s + (0.866 + 0.5i)9-s + (−0.5 − 0.866i)10-s + (0.965 + 0.258i)11-s + (−0.258 − 0.965i)12-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)15-s + (−0.5 + 0.866i)16-s + (0.258 − 0.965i)17-s + (−0.5 − 0.866i)18-s + (−0.965 + 0.258i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.740 - 0.672i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.740 - 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.120729259 - 0.4331868801i\)
\(L(\frac12)\) \(\approx\) \(1.120729259 - 0.4331868801i\)
\(L(1)\) \(\approx\) \(0.7313061811 - 0.1652243877i\)
\(L(1)\) \(\approx\) \(0.7313061811 - 0.1652243877i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.965 + 0.258i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (0.258 - 0.965i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.965 + 0.258i)T \)
53 \( 1 + (0.965 + 0.258i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (0.258 - 0.965i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.258 - 0.965i)T \)
83 \( 1 + T \)
89 \( 1 + (0.258 + 0.965i)T \)
97 \( 1 + (-0.707 - 0.707i)T \)
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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

−25.52860536691427665647329450791, −24.48830012525072604299632401216, −23.820874238117204566286245281993, −22.92350170250253541935295159031, −21.56903456017482174651040389988, −21.13603626701699774564018860867, −19.744510428967056279896637203356, −18.81944007348130141684605366757, −17.794299539719807625989161220231, −17.0585379024266063709004197262, −16.651523284238791344510164114576, −15.60336068237233864010450126769, −14.53357465989634505271846586877, −13.376511560459599674433402115296, −12.10134556032672846007235413475, −11.08791758557674040586566101347, −10.25378585084302810349187153195, −9.25950980967308047302030501721, −8.561498092592958058105981635783, −6.86487300941162102124571611134, −6.20448452856461180650842635485, −5.38286479745329088300275437347, −4.0920065382855112857869064001, −1.80459402895669459046604698495, −0.93031906449193092244548351861, 0.76998338992666838948907086804, 1.81113530361161868829334646311, 3.13502173703753521092464745832, 4.72352486671970834418803160614, 6.30794587943809912218963565321, 6.70303943765285184761322406029, 8.06887064106047099641722589391, 9.336052999374381801514651819, 10.27217466076381070840611108300, 10.92139244996304725104868340407, 11.93041333003061814529359230162, 12.794997522112478108144264027242, 13.84098702650594485681333115575, 15.29094415125063144029827889109, 16.485332799991637628635876739900, 17.19294919584169560718250700808, 17.93207711641028792816799666134, 18.57426294895962592546502968070, 19.5054237868513468291223014716, 20.80358061601632592123132616495, 21.46632632942509653416113163858, 22.52341199220091634645520334138, 23.01845756103401711067958737664, 24.82342805459802650374595258914, 25.047974643999046936761850558113

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