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Properties

Label 1-287-287.13-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: $\chi_{287} (13, \cdot )$
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

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

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