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Properties

Label 1-1287-1287.1165-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.919 + 0.393i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + (−0.866 − 0.5i)7-s + i·8-s + 10-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s − i·19-s + (0.866 + 0.5i)20-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s − i·28-s + (0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 − 0.5i)5-s + (−0.866 − 0.5i)7-s + i·8-s + 10-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 17-s − i·19-s + (0.866 + 0.5i)20-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s − i·28-s + (0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.919 + 0.393i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.919 + 0.393i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.903528146 + 0.5948196807i\)
\(L(\frac12)\) \(\approx\) \(2.903528146 + 0.5948196807i\)
\(L(1)\) \(\approx\) \(1.878256817 + 0.3757867482i\)
\(L(1)\) \(\approx\) \(1.878256817 + 0.3757867482i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + T \)
19 \( 1 - iT \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + iT \)
73 \( 1 + iT \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−21.0303797529227550058288913080, −20.47091226218037651940465935948, −19.40646603503610436974578536054, −18.71842675718686326177304781283, −18.32830368982449733125043961350, −16.96145577849816812237435349576, −16.33699975433919122709240679055, −15.316880018558498961023570309604, −14.703286739750403097019806558930, −13.898140986628326843742071817511, −13.33619820283245312205790674996, −12.29114213046317433079698541373, −12.055968869742081958272833512748, −10.61471646625811431293669554110, −10.25042133414568999981284623286, −9.517740303165233745103341786239, −8.49896018837375256159132549741, −7.01078281070101375760671569872, −6.46248029025939569540490351963, −5.65351930481542734034617611728, −5.0157000861785698456131571757, −3.64273291599713852736877966305, −3.02406067461667841939631160778, −2.216440518989055992118098910754, −1.15881219374023681021102013194, 1.01683667606599724786470492580, 2.392031809371014337779244617917, 3.18074880189681433484152689072, 4.18124619553936404332674716884, 5.06272336386867461758029749432, 5.85068802611362554478888792615, 6.55119323524425149593406885170, 7.36475247492560140906146009925, 8.30617004702039386312246532895, 9.348631571721878430332841192583, 9.990086507659113537092933677142, 11.05923984285020407123017608646, 12.04866738108567657938145448038, 12.78740217626358474049416168761, 13.511780867045778201525448085075, 13.83458006991210699695657254057, 14.893996733192976694312390874, 15.73662746752934520633732559329, 16.40941067931280977872727281134, 17.170461049524788123177837991809, 17.5554879112840260014090801182, 18.818114695318026569679004763999, 19.792599364546979239352679927133, 20.42017028991677682554570146418, 21.42269251212358920090489864056

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