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Properties

Label 1-287-287.12-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: $\chi_{287} (12, \cdot )$
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

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

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