↑

Properties

Label 1-1287-1287.1238-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.448 + 0.893i$
Analytic cond. $5.97680$
Root an. cond. $5.97680$
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  + (0.913 + 0.406i)2-s + (0.669 + 0.743i)4-s + (0.104 − 0.994i)5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (0.5 − 0.866i)10-s + (−0.669 + 0.743i)14-s + (−0.104 + 0.994i)16-s + (−0.104 + 0.994i)17-s + (−0.669 + 0.743i)19-s + (0.809 − 0.587i)20-s − 23-s + (−0.978 − 0.207i)25-s + (−0.913 + 0.406i)28-s + (−0.978 + 0.207i)29-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (0.669 + 0.743i)4-s + (0.104 − 0.994i)5-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (0.5 − 0.866i)10-s + (−0.669 + 0.743i)14-s + (−0.104 + 0.994i)16-s + (−0.104 + 0.994i)17-s + (−0.669 + 0.743i)19-s + (0.809 − 0.587i)20-s − 23-s + (−0.978 − 0.207i)25-s + (−0.913 + 0.406i)28-s + (−0.978 + 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.448 + 0.893i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.448 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.141110492 + 1.848591578i\)
\(L(\frac12)\) \(\approx\) \(1.141110492 + 1.848591578i\)
\(L(1)\) \(\approx\) \(1.469267422 + 0.6758564902i\)
\(L(1)\) \(\approx\) \(1.469267422 + 0.6758564902i\)

Euler product

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

−20.73113034359120876354720641853, −20.13197683927996053782566560437, −19.37269636847003328323291559314, −18.70002300750414452960448072907, −17.81370797083516681990513715383, −16.853224423797660878921413536406, −15.93981326571921493133599645867, −15.23402681248175753316541274670, −14.44962338111546093838555462076, −13.68806647920898672167240543669, −13.34673579644183960083222718577, −12.21214470516903835426109200144, −11.377921351397472877433658082195, −10.75920211674551455193487622625, −10.07744573151983935128160447623, −9.327594163868257349142642705202, −7.725444836170916646822826094737, −7.04257899932172653706723042653, −6.394495059571527887976464451911, −5.51442480891222087018109398018, −4.331828571760400971683070299790, −3.76682305771058163792260363243, −2.75799168343571261011443726970, −2.070284677331708632378138548026, −0.57006969680242244013984985539, 1.60684285048835394246516765733, 2.39763629728646717054192943898, 3.608199728093550760529446328629, 4.35488646566914162947264035957, 5.29716934754551926980303655267, 5.9623948291731898056997380814, 6.58710468959218205296091809281, 8.0822327558609131570244464403, 8.304811584885225208082541058521, 9.36931915899091358894219705794, 10.3566821562698824492341393045, 11.57162577031384819800647036041, 12.172989815122512831072810956348, 12.830759564952141427936402507129, 13.35693404530127145252558231434, 14.42804702338411889367968938805, 15.13344947195385119175021786919, 15.82018269015781966750156695349, 16.56593484629837345492593706026, 17.12487088619066874364676096065, 18.07495838882926385146928965782, 19.106490573924670077551170875171, 19.96380115511381823327807041068, 20.630132381254260889817949796602, 21.56937497009915661658556033671

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