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Properties

Label 1-1287-1287.1109-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.196 + 0.980i$
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

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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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.196 + 0.980i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.196 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6731320171 - 0.8217710677i\)
\(L(\frac12)\) \(\approx\) \(-0.6731320171 - 0.8217710677i\)
\(L(1)\) \(\approx\) \(1.118909628 - 0.8798763445i\)
\(L(1)\) \(\approx\) \(1.118909628 - 0.8798763445i\)

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 \( 1 + (0.809 + 0.587i)T \)
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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

−21.55376620471565556448206474496, −20.92521757461285107854148828726, −19.855592884399987626391993520392, −19.111153923945436904993145879530, −18.21298007591100098317009814443, −17.67056617931300283070788553653, −16.34734295102941895561961632459, −15.976449257389745232627724921925, −15.06060975908179718088757574719, −14.532088728311822993448446028, −13.84755633038628364107393471575, −12.89957026200257001463282074381, −12.07121063830676725384527105648, −11.55814619906861207407178782622, −10.6129992697415986470720246396, −9.695964002424755286592225072395, −8.561506977543479780494293798671, −7.724599466277826356485274433, −6.89975379223886419998841271637, −6.105067103921529348765207441924, −5.55299881467082240816982572437, −4.38429562036525399039307544344, −3.49124408437832362020885071824, −2.67149706715758369741110483012, −1.99189268565401899190033595436, 0.13752634067714946898436966309, 1.15504173206387657469117188499, 2.02321346789840870901589825072, 3.32203281107450752470426325336, 4.12813546048377494952998752410, 4.680636551560551477208397917394, 5.7004690942572809185744601618, 6.50851936055492722718744883802, 7.423781132686952393290500862502, 8.42776497160628076180613398659, 9.410908438415123992630547103768, 10.36672251574288359514691191677, 10.86292346241664343638912484671, 12.026433156247853336910317022149, 12.544052881510326635950089807267, 13.30432266467960254917218270190, 13.86236296113177426642341480846, 14.76636398462469804096626381686, 15.69174640963076153858958317464, 16.304152312653282358898523869032, 17.054162278227883941756852989634, 17.88575100884416790912780311433, 19.37226865009547364251828714098, 19.55602968382710561970007918199, 20.34732156657427007240779738302

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