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Properties

Label 1-287-287.62-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.597 + 0.801i$
Analytic cond. $30.8424$
Root an. cond. $30.8424$
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  + (0.809 + 0.587i)2-s − i·3-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)5-s + (0.587 − 0.809i)6-s + (−0.309 + 0.951i)8-s − 9-s + (−0.309 + 0.951i)10-s + (0.951 + 0.309i)11-s + (0.951 − 0.309i)12-s + (−0.587 + 0.809i)13-s + (0.951 − 0.309i)15-s + (−0.809 + 0.587i)16-s + (0.951 + 0.309i)17-s + (−0.809 − 0.587i)18-s + (−0.587 − 0.809i)19-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)2-s − i·3-s + (0.309 + 0.951i)4-s + (0.309 + 0.951i)5-s + (0.587 − 0.809i)6-s + (−0.309 + 0.951i)8-s − 9-s + (−0.309 + 0.951i)10-s + (0.951 + 0.309i)11-s + (0.951 − 0.309i)12-s + (−0.587 + 0.809i)13-s + (0.951 − 0.309i)15-s + (−0.809 + 0.587i)16-s + (0.951 + 0.309i)17-s + (−0.809 − 0.587i)18-s + (−0.587 − 0.809i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.597 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.597 + 0.801i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.597 + 0.801i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.141644013 + 2.274593254i\)
\(L(\frac12)\) \(\approx\) \(1.141644013 + 2.274593254i\)
\(L(1)\) \(\approx\) \(1.432992952 + 0.7116754499i\)
\(L(1)\) \(\approx\) \(1.432992952 + 0.7116754499i\)

Euler product

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

−24.91600022417472751992321003093, −23.984995366490885215851225360834, −22.88582147238184157299392663704, −22.15580038882540975268202736246, −21.36515315870187493362426455058, −20.57618768692326427633025758499, −19.99749339729105184894490904397, −19.00236055492617177897605732962, −17.38094210522645581075479225472, −16.58902976563885755190263755896, −15.685072953240910337678638750754, −14.627239519773701357177452549950, −13.99203316233788016855197899671, −12.72473688416213117893893659747, −11.974578625873890282363837491200, −10.955422195471779397351697821013, −9.79640104725038645117149757749, −9.3567232359575246675151573966, −7.921380653455999103863056714013, −5.936830135998335129931015194873, −5.457433239984407220307461013725, −4.261364687219140583850687048, −3.53600899709730534628806542507, −2.054373147587346110476915474, −0.54820969012492763508589422875, 1.803450010259889679045869974957, 2.81080120531760334844177050658, 4.04876805207511976713955884234, 5.53393270284680412127852241343, 6.599223341619843417491063402, 6.98487917906612466404216270506, 8.079753804588672218662795669292, 9.35338379672294219065542514469, 10.965609766117929871982413555242, 11.91269654479156499729373558312, 12.635712243886909950159779262146, 13.832380699332135010902249382440, 14.39204593215702307365265984493, 15.048852605402115610114686326561, 16.638772295225048727405631507075, 17.279914332488479571031576291, 18.221680318449168170800519503608, 19.19968512432228601542127792183, 20.13002354232977206925518852392, 21.50942360114922232511790388521, 22.19925069866292575644375149790, 23.00157119098792159614303619638, 23.86574220189186535416256506367, 24.57465077813718454429345343922, 25.66981672933698464325023381044

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