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Properties

Label 1-1287-1287.1280-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.654 - 0.756i$
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  + (0.207 − 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.743 + 0.669i)5-s + (−0.587 − 0.809i)7-s + (−0.587 + 0.809i)8-s + (0.5 + 0.866i)10-s + (−0.913 + 0.406i)14-s + (0.669 + 0.743i)16-s + (0.669 + 0.743i)17-s + (−0.406 − 0.913i)19-s + (0.951 − 0.309i)20-s + 23-s + (0.104 − 0.994i)25-s + (0.207 + 0.978i)28-s + (0.104 + 0.994i)29-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.743 + 0.669i)5-s + (−0.587 − 0.809i)7-s + (−0.587 + 0.809i)8-s + (0.5 + 0.866i)10-s + (−0.913 + 0.406i)14-s + (0.669 + 0.743i)16-s + (0.669 + 0.743i)17-s + (−0.406 − 0.913i)19-s + (0.951 − 0.309i)20-s + 23-s + (0.104 − 0.994i)25-s + (0.207 + 0.978i)28-s + (0.104 + 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.654 - 0.756i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1280, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.654 - 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3780165661 - 0.8266174723i\)
\(L(\frac12)\) \(\approx\) \(0.3780165661 - 0.8266174723i\)
\(L(1)\) \(\approx\) \(0.7165269501 - 0.4331698883i\)
\(L(1)\) \(\approx\) \(0.7165269501 - 0.4331698883i\)

Euler product

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

−21.25810597172075825476251363790, −20.76591384574645908724494115366, −19.5219693429226009597291778754, −18.882893295211094167890138848212, −18.3146962702809980041498819666, −17.14070203227296770660596158799, −16.556199579114716656095614235581, −15.96111646469440628660671178943, −15.220222140703179646286595048180, −14.66629116804369433493867199003, −13.550158276897714251419347031154, −12.82709728178739172640890106154, −12.20383948098543987097844676467, −11.50156222688174013405303743620, −10.03602322683529683883927148732, −9.241951553167904483559447122798, −8.61680237523771171939523323914, −7.78919767607469028682053367222, −7.09316595035588250435218386681, −5.95833414276697815587372164190, −5.4329933806552193793036151210, −4.431497111911793245922483063746, −3.65828753199163724884263625447, −2.65129181139835011146959801268, −0.91376537644780544951696524381, 0.449293405461773065723715692413, 1.645994160337248833124234563979, 3.03126438835014494163878150147, 3.386684957321155526501416601782, 4.314942285049706360587907372251, 5.182620302782592349444244741146, 6.47790900508201429295368660650, 7.144883007804175173300684411190, 8.22852437669152919016393274743, 9.063006169679690318640661796800, 10.11511667680282083193455482542, 10.6773035470042148635599763545, 11.22885095404801184867751975694, 12.25710640206297599109580219991, 12.84052085595600415969144847051, 13.699160242437338256012499706863, 14.48218175307341531367208124205, 15.14737557143120836777533573143, 16.10027644371100521085557967401, 17.104185589491757704423201375867, 17.79674659492603292980814779601, 18.921673621356052788616323621899, 19.1787634479708013622564597970, 19.94107023901873992370227484893, 20.565100738080205723835039042957

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