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Properties

Label 1-287-287.89-r0-0-0
Degree $1$
Conductor $287$
Sign $0.998 - 0.0577i$
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  + (0.994 + 0.104i)2-s + (−0.965 + 0.258i)3-s + (0.978 + 0.207i)4-s + (−0.743 − 0.669i)5-s + (−0.987 + 0.156i)6-s + (0.951 + 0.309i)8-s + (0.866 − 0.5i)9-s + (−0.669 − 0.743i)10-s + (−0.0523 + 0.998i)11-s + (−0.998 + 0.0523i)12-s + (−0.156 − 0.987i)13-s + (0.891 + 0.453i)15-s + (0.913 + 0.406i)16-s + (0.998 + 0.0523i)17-s + (0.913 − 0.406i)18-s + (0.933 − 0.358i)19-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (−0.965 + 0.258i)3-s + (0.978 + 0.207i)4-s + (−0.743 − 0.669i)5-s + (−0.987 + 0.156i)6-s + (0.951 + 0.309i)8-s + (0.866 − 0.5i)9-s + (−0.669 − 0.743i)10-s + (−0.0523 + 0.998i)11-s + (−0.998 + 0.0523i)12-s + (−0.156 − 0.987i)13-s + (0.891 + 0.453i)15-s + (0.913 + 0.406i)16-s + (0.998 + 0.0523i)17-s + (0.913 − 0.406i)18-s + (0.933 − 0.358i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.998 - 0.0577i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.998 - 0.0577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.622000511 - 0.04688508253i\)
\(L(\frac12)\) \(\approx\) \(1.622000511 - 0.04688508253i\)
\(L(1)\) \(\approx\) \(1.386364537 + 0.02741848644i\)
\(L(1)\) \(\approx\) \(1.386364537 + 0.02741848644i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.994 + 0.104i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (-0.743 - 0.669i)T \)
11 \( 1 + (-0.0523 + 0.998i)T \)
13 \( 1 + (-0.156 - 0.987i)T \)
17 \( 1 + (0.998 + 0.0523i)T \)
19 \( 1 + (0.933 - 0.358i)T \)
23 \( 1 + (0.104 - 0.994i)T \)
29 \( 1 + (0.453 - 0.891i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (0.629 + 0.777i)T \)
53 \( 1 + (-0.544 - 0.838i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (-0.406 + 0.913i)T \)
67 \( 1 + (0.838 - 0.544i)T \)
71 \( 1 + (-0.891 + 0.453i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.258 - 0.965i)T \)
83 \( 1 - T \)
89 \( 1 + (0.358 + 0.933i)T \)
97 \( 1 + (-0.891 - 0.453i)T \)
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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

−25.23842267009658503789125252347, −24.16777750211544614228005074088, −23.58770013182228955125653607827, −22.97756016436383104174481957823, −21.915582877360847527775902411671, −21.558605266926694882496970988350, −20.1730145108893217913623464646, −18.96384497187720663609600574514, −18.64695635365297081930847393567, −16.98291743069263345282432992027, −16.23147194361217423481874560977, −15.480556564501637006307627796119, −14.25414084486511600713682920477, −13.54825875373635366730037124173, −12.173104595842625609527249730189, −11.684092173018255234679125209576, −10.95968123910069770378453808335, −9.92042926321809187415987878084, −7.88796285225178967412162064625, −7.052092422936664061095345834263, −6.12696381414423915686373580151, −5.18569135758509838034053143578, −3.98677852339096513444837769322, −2.99359216714770906258860714976, −1.30995574010842157585758758629, 1.126518647323937927345679892089, 3.03203535980231777138524907541, 4.3338783823680398417352384240, 4.950595556624896497108592178704, 5.90137023744126427572657096861, 7.16586812525074199031007184863, 7.97388126284451550832341184584, 9.76276015850216517056439856923, 10.73154342401614266346130609096, 11.87158194191169586174770837796, 12.34548526190749584735767211650, 13.121973673352337374123832392130, 14.63523879110876364077010836529, 15.498881210197662252869005275, 16.13571858731196411950936511396, 17.02190018503422429338120452843, 17.96151170586485719151301129080, 19.44154834306873577864948122196, 20.4323642860078248378256594427, 21.018975336878893713072854968970, 22.19908804128718125656301248063, 23.023022554677498245521095593441, 23.32624046354107681475577066809, 24.48271070654523321592286585784, 25.05523378691565642511852461236

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