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Properties

Label 1-287-287.265-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.999 + 0.0130i$
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  + (0.587 − 0.809i)2-s + (0.707 − 0.707i)3-s + (−0.309 − 0.951i)4-s + (−0.951 + 0.309i)5-s + (−0.156 − 0.987i)6-s + (−0.951 − 0.309i)8-s − i·9-s + (−0.309 + 0.951i)10-s + (−0.453 − 0.891i)11-s + (−0.891 − 0.453i)12-s + (−0.987 + 0.156i)13-s + (−0.453 + 0.891i)15-s + (−0.809 + 0.587i)16-s + (0.891 − 0.453i)17-s + (−0.809 − 0.587i)18-s + (−0.987 − 0.156i)19-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)2-s + (0.707 − 0.707i)3-s + (−0.309 − 0.951i)4-s + (−0.951 + 0.309i)5-s + (−0.156 − 0.987i)6-s + (−0.951 − 0.309i)8-s − i·9-s + (−0.309 + 0.951i)10-s + (−0.453 − 0.891i)11-s + (−0.891 − 0.453i)12-s + (−0.987 + 0.156i)13-s + (−0.453 + 0.891i)15-s + (−0.809 + 0.587i)16-s + (0.891 − 0.453i)17-s + (−0.809 − 0.587i)18-s + (−0.987 − 0.156i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.999 + 0.0130i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (265, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.999 + 0.0130i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.008642111241 - 1.328257878i\)
\(L(\frac12)\) \(\approx\) \(0.008642111241 - 1.328257878i\)
\(L(1)\) \(\approx\) \(0.7940055176 - 0.9700111974i\)
\(L(1)\) \(\approx\) \(0.7940055176 - 0.9700111974i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.587 - 0.809i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (-0.453 - 0.891i)T \)
13 \( 1 + (-0.987 + 0.156i)T \)
17 \( 1 + (0.891 - 0.453i)T \)
19 \( 1 + (-0.987 - 0.156i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (-0.891 - 0.453i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (-0.156 - 0.987i)T \)
53 \( 1 + (0.891 + 0.453i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.587 + 0.809i)T \)
67 \( 1 + (0.453 - 0.891i)T \)
71 \( 1 + (0.453 + 0.891i)T \)
73 \( 1 - iT \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.156 + 0.987i)T \)
97 \( 1 + (0.453 - 0.891i)T \)
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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

−25.88834163691653806136009954944, −25.11731005069706119712297238487, −24.22840867072095786806059620983, −23.22250699050873297301666687979, −22.59671070703684680749019049213, −21.42568868970948074632238012196, −20.7652889086323602704953617594, −19.79050753116100222170026372544, −18.855432707427615340324127475782, −17.35853864652831988318015542897, −16.51604295727639185131522029502, −15.73028718537892727953551807161, −14.73869642299546226526042506735, −14.59296610473835758388970868155, −12.8572315502864716230626510677, −12.4857267932096020373545213088, −10.99335769440097109864335299530, −9.738274927579125764720394105757, −8.624518800881915441705389837311, −7.82963730507770706029270359501, −7.05916258256848279573879113407, −5.292145449754201390352925477846, −4.55832808931091398665779734973, −3.66436738218580058834733614226, −2.51821904458161461643065720924, 0.65804161263535462717635468875, 2.33164649652259975019503548550, 3.160881980631636481113191394180, 4.13072965590479801636500402145, 5.5356128119397366702463058118, 6.86768227542550070597275985436, 7.85999240621204090449062034030, 8.934190680925228163720213978761, 10.09324699188859677744883589184, 11.32650505824298471036845399994, 11.9999532117630657644201605378, 12.97008214061010247574255882493, 13.78572386770200722084142804408, 14.835047060269186635427059023847, 15.299262135967732739024150022178, 16.85725497864653202777199584714, 18.38095447449574697015835067055, 19.063419293774406361726642274921, 19.44603556299523081753938200236, 20.51946349396067471244446762376, 21.28795841168241233019310677706, 22.40968312971091884851667814296, 23.40802151030703260490456469927, 23.944118214775109440506826924524, 24.76561460228509877745198910229

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