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Properties

Label 1-287-287.208-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.447 - 0.894i$
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  + (−0.866 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01226227318 + 0.01984133084i\)
\(L(\frac12)\) \(\approx\) \(0.01226227318 + 0.01984133084i\)
\(L(1)\) \(\approx\) \(0.3333667721 + 0.1446915858i\)
\(L(1)\) \(\approx\) \(0.3333667721 + 0.1446915858i\)

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

−24.90172275506878052712132197022, −24.00470863151254896080858633995, −23.11244082062333293093279765170, −22.28487453431858969861295317175, −20.93390615761376512241427901206, −20.514583402873210469648400858321, −19.15146205765653365357757866945, −18.582049248293927739332849266926, −17.772147612066504946699445775482, −16.5795064385710594549570403585, −16.206138545923586389771545450241, −15.20928693922305878485022799121, −13.04445083878913221507627776289, −12.734979569204165519435078557460, −11.5142812385336575403328953002, −10.97345444566247165599328291389, −10.02270942302608450550162825950, −8.595901725092712034516363541141, −7.8383708395713958163423720520, −6.873232432094527818538839812135, −5.494459677967396391258084154431, −4.27210568429108453648650166559, −2.90612382298349875133333124737, −1.242209201048576649150659508853, −0.02590511278469145361725472932, 1.79253259356975034642461396051, 3.69788617617603309768560669971, 4.99336802346453631584549990305, 6.126129627813957488637144629519, 7.00038771937304903861324259629, 7.92179478337181935824339192449, 9.08103110167198077850267673433, 10.40339368592682220192992295821, 10.88967325162288653066325274122, 11.75171963278578494194502562771, 12.98739026359079482693702275543, 14.62301369632340745145484736678, 15.48504813482587939735044526246, 16.01972637568541995239003847618, 17.025279982018294611501839795124, 17.8598234600102848787311053319, 18.79000787957408772824203457934, 19.32762423506000270110406246451, 20.7203043175833062309941929399, 21.598474120250760005492506154884, 22.98179317762476418465840035336, 23.55646679745986085767492416468, 23.98763591231882176777071393866, 25.52234848400087567960453407400, 26.314751824583090671799469103837

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