↑

Properties

Label 1-1287-1287.400-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.982 + 0.184i$
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  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (0.978 − 0.207i)5-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.5 + 0.866i)10-s + (−0.104 + 0.994i)14-s + (−0.978 + 0.207i)16-s + (−0.978 + 0.207i)17-s + (0.104 − 0.994i)19-s + (−0.309 − 0.951i)20-s + 23-s + (0.913 − 0.406i)25-s + (−0.669 − 0.743i)28-s + (0.913 + 0.406i)29-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (0.978 − 0.207i)5-s + (0.809 − 0.587i)7-s + (0.809 + 0.587i)8-s + (−0.5 + 0.866i)10-s + (−0.104 + 0.994i)14-s + (−0.978 + 0.207i)16-s + (−0.978 + 0.207i)17-s + (0.104 − 0.994i)19-s + (−0.309 − 0.951i)20-s + 23-s + (0.913 − 0.406i)25-s + (−0.669 − 0.743i)28-s + (0.913 + 0.406i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.982 + 0.184i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (400, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.982 + 0.184i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.484398306 + 0.1380615018i\)
\(L(\frac12)\) \(\approx\) \(1.484398306 + 0.1380615018i\)
\(L(1)\) \(\approx\) \(1.018614774 + 0.1559269219i\)
\(L(1)\) \(\approx\) \(1.018614774 + 0.1559269219i\)

Euler product

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

−21.16063883941404054480201047245, −20.34029704037633526308606975740, −19.3567253505428953951079716481, −18.6142989379007611294269819318, −17.92664807582204811363170876129, −17.52345426944391880349560259756, −16.68380396427915693775662948973, −15.76763967788961372704471278760, −14.70622473921373149312914131488, −13.993111394798506523042488772259, −13.07698452118384616637849040391, −12.4380855491156189574464088918, −11.40129998234673978704914503010, −10.9135473166928350038565544294, −10.02194535429353619533440341330, −9.230182163912806710508840219345, −8.647548052419013427559510012576, −7.72221248672342376624363542330, −6.784619049425858019794892999480, −5.7401911265045216062285168816, −4.83226225959518788503448509769, −3.78460489843764307790412502166, −2.51646361564560506666141740703, −2.10600038598726564443941665881, −1.05048245579404485185033494273, 0.92290932038021602208084564608, 1.69753100355775057001174342470, 2.78108198506620937440893711252, 4.5885570083507401413629836831, 4.9087507459864133343461847262, 6.011076858873853008794953405245, 6.79309048879851415973787643047, 7.48366227800050118820538014531, 8.62009388675084072863076627654, 9.01612431662600541970753795168, 9.9882731134797220859670006539, 10.78180060841087920866622442156, 11.293365148377110718321884980168, 12.79802655834632522673035650489, 13.56158288366320911372423460858, 14.18354963818275933218636375569, 14.91461602802319294792452120160, 15.75704450833160651686579342077, 16.66392635105746243899926912538, 17.32531840171369120407458665451, 17.80758272164055434169319130980, 18.372638790360374056798986530, 19.57522202630839375829466840968, 20.07655875793525785787296749954, 20.99259374793142026039702838146

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