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Properties

Label 1-287-287.24-r0-0-0
Degree $1$
Conductor $287$
Sign $0.143 - 0.989i$
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  + (0.207 − 0.978i)2-s + (0.965 + 0.258i)3-s + (−0.913 − 0.406i)4-s + (0.994 − 0.104i)5-s + (0.453 − 0.891i)6-s + (−0.587 + 0.809i)8-s + (0.866 + 0.5i)9-s + (0.104 − 0.994i)10-s + (−0.629 − 0.777i)11-s + (−0.777 − 0.629i)12-s + (−0.891 − 0.453i)13-s + (0.987 + 0.156i)15-s + (0.669 + 0.743i)16-s + (0.777 − 0.629i)17-s + (0.669 − 0.743i)18-s + (0.0523 − 0.998i)19-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)2-s + (0.965 + 0.258i)3-s + (−0.913 − 0.406i)4-s + (0.994 − 0.104i)5-s + (0.453 − 0.891i)6-s + (−0.587 + 0.809i)8-s + (0.866 + 0.5i)9-s + (0.104 − 0.994i)10-s + (−0.629 − 0.777i)11-s + (−0.777 − 0.629i)12-s + (−0.891 − 0.453i)13-s + (0.987 + 0.156i)15-s + (0.669 + 0.743i)16-s + (0.777 − 0.629i)17-s + (0.669 − 0.743i)18-s + (0.0523 − 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.143 - 0.989i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.143 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.475314674 - 1.276666729i\)
\(L(\frac12)\) \(\approx\) \(1.475314674 - 1.276666729i\)
\(L(1)\) \(\approx\) \(1.401264044 - 0.7679671170i\)
\(L(1)\) \(\approx\) \(1.401264044 - 0.7679671170i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.207 - 0.978i)T \)
3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (0.994 - 0.104i)T \)
11 \( 1 + (-0.629 - 0.777i)T \)
13 \( 1 + (-0.891 - 0.453i)T \)
17 \( 1 + (0.777 - 0.629i)T \)
19 \( 1 + (0.0523 - 0.998i)T \)
23 \( 1 + (0.978 + 0.207i)T \)
29 \( 1 + (-0.156 + 0.987i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (0.544 + 0.838i)T \)
53 \( 1 + (-0.933 + 0.358i)T \)
59 \( 1 + (-0.669 + 0.743i)T \)
61 \( 1 + (0.743 - 0.669i)T \)
67 \( 1 + (0.358 + 0.933i)T \)
71 \( 1 + (-0.987 + 0.156i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.258 - 0.965i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.998 - 0.0523i)T \)
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

−25.531146223709073948157009241756, −25.04574842691323177403142213748, −24.1845305558995984481508749491, −23.24530011098577544710895344714, −22.152920171574547336701498431633, −21.205533306563605617415823465711, −20.58352344588688576703147721438, −19.000058595244290398730755492520, −18.49193574407361302707237918529, −17.34921624524319705608543968315, −16.68276784215170380185673225543, −15.19980808074681360395338787159, −14.74797173176336255080240535170, −13.84887554658019343719201068364, −13.03610075069161882081830546945, −12.27914282731788295198868946638, −10.02064054758170983160521351689, −9.66111124571035972140008145088, −8.42330916287130644951915429635, −7.53240049721798761105837012081, −6.645341723334078560123619396364, −5.48900311853134785536336152994, −4.36015435308753534675368976664, −2.998110318793723991085271108658, −1.7733733997279216209591572248, 1.29579085839499901878280220855, 2.69131566684462745541947175926, 3.111441343045951072868410160526, 4.804568092297929282701949686488, 5.46085299878792493894350665516, 7.26896479948833981744019962982, 8.634672339169411850076796203216, 9.34754205233796207836402818857, 10.19636799404961689421305230959, 11.00466320354175738493248588026, 12.570676272343258963599193019836, 13.223723764518325557987965504095, 14.08001908265510402890401156575, 14.72723845632925206388145977523, 16.03371287757256815214175973916, 17.33974110394565739958808475024, 18.31571086793819350926135684052, 19.14049612710797116452946399656, 20.05086566282701155430865510266, 20.84564032393410437602267984049, 21.5855594568772889286778061414, 22.084984234049953557740142267570, 23.45965385792317125560964773476, 24.52937645811218456435067645326, 25.34478143708158173232402869294

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