Properties

Label 1-1287-1287.31-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.217 + 0.976i$
Analytic cond. $138.307$
Root an. cond. $138.307$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.406 − 0.913i)2-s + (−0.669 + 0.743i)4-s + (0.406 − 0.913i)5-s + (0.207 − 0.978i)7-s + (0.951 + 0.309i)8-s − 10-s + (−0.978 + 0.207i)14-s + (−0.104 − 0.994i)16-s + (0.809 − 0.587i)17-s + (−0.951 − 0.309i)19-s + (0.406 + 0.913i)20-s + (0.5 + 0.866i)23-s + (−0.669 − 0.743i)25-s + (0.587 + 0.809i)28-s + (−0.978 − 0.207i)29-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (−0.669 + 0.743i)4-s + (0.406 − 0.913i)5-s + (0.207 − 0.978i)7-s + (0.951 + 0.309i)8-s − 10-s + (−0.978 + 0.207i)14-s + (−0.104 − 0.994i)16-s + (0.809 − 0.587i)17-s + (−0.951 − 0.309i)19-s + (0.406 + 0.913i)20-s + (0.5 + 0.866i)23-s + (−0.669 − 0.743i)25-s + (0.587 + 0.809i)28-s + (−0.978 − 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.217 + 0.976i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.217 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2256702612 - 0.1810122770i\)
\(L(\frac12)\) \(\approx\) \(-0.2256702612 - 0.1810122770i\)
\(L(1)\) \(\approx\) \(0.5985701797 - 0.5095974114i\)
\(L(1)\) \(\approx\) \(0.5985701797 - 0.5095974114i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.406 - 0.913i)T \)
5 \( 1 + (0.406 - 0.913i)T \)
7 \( 1 + (0.207 - 0.978i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (0.994 + 0.104i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (0.207 + 0.978i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.743 + 0.669i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.743 + 0.669i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (0.913 - 0.406i)T \)
83 \( 1 + (-0.994 + 0.104i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.406 - 0.913i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.448681531458831533351188215000, −20.882368023734421610400812878575, −19.34456714679978550092410944394, −18.979557620954657115821017231592, −18.35962013994452384331091744256, −17.57665578113585261346718131811, −16.95061762759907657549693287432, −16.03250170830595502660729722657, −15.08420027814391978988003272348, −14.77462992450798837979188523374, −14.07157359382536334537527118027, −13.05742490410518184641661808485, −12.21840025033946778220241021485, −11.00429855820317378091716613804, −10.401963002133749121669789381122, −9.560307912795824905248236191154, −8.73062427461470445289794355424, −8.01502865848235607343944861742, −7.08649556686080020121189996207, −6.222409683830166372505164332, −5.7464454751754595071741876630, −4.77740038315607720847052714782, −3.59209131682678844446206834411, −2.40741695576243230268730013465, −1.508758042803275880634655015704, 0.06985841825568149509359287737, 1.04192524827818458367252226878, 1.70959032903450904485831274950, 2.91145395557019846744008190953, 3.93127540249667080461238406981, 4.67548850823878200790001291445, 5.468617911599705104051754096080, 6.858467213754622041739891355409, 7.80530393821756830263498533224, 8.46866585337192711041206402958, 9.44172467801917310411671071027, 9.93755415817451252511036313446, 10.84391643739691123196993180028, 11.594468233092452052912866742554, 12.44684333552529959913256541287, 13.23328129039525977322172906362, 13.68649069291456151440426789569, 14.59621924182677031346165146085, 15.92453888346908909630410967546, 16.72496006822623192084765753830, 17.29223922558426805019716375168, 17.762961087736715102089086232554, 18.95213607721178644029061503464, 19.47892635466375584142670846486, 20.367818482538482713535204804106

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