Properties

Label 1-1287-1287.526-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.523 + 0.851i$
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  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.406 − 0.913i)5-s + (0.743 + 0.669i)7-s + (0.951 + 0.309i)8-s + (0.5 + 0.866i)10-s + (−0.978 + 0.207i)14-s + (−0.809 + 0.587i)16-s + (0.104 + 0.994i)17-s + (0.743 − 0.669i)19-s + (−0.994 − 0.104i)20-s + (0.5 − 0.866i)23-s + (−0.669 − 0.743i)25-s + (0.406 − 0.913i)28-s + (0.309 + 0.951i)29-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (0.406 − 0.913i)5-s + (0.743 + 0.669i)7-s + (0.951 + 0.309i)8-s + (0.5 + 0.866i)10-s + (−0.978 + 0.207i)14-s + (−0.809 + 0.587i)16-s + (0.104 + 0.994i)17-s + (0.743 − 0.669i)19-s + (−0.994 − 0.104i)20-s + (0.5 − 0.866i)23-s + (−0.669 − 0.743i)25-s + (0.406 − 0.913i)28-s + (0.309 + 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.523 + 0.851i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (526, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.523 + 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.769083450 + 0.9888983474i\)
\(L(\frac12)\) \(\approx\) \(1.769083450 + 0.9888983474i\)
\(L(1)\) \(\approx\) \(0.9750541353 + 0.2809351450i\)
\(L(1)\) \(\approx\) \(0.9750541353 + 0.2809351450i\)

Euler product

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

−20.76186277112398820809012298945, −19.93755188337665670904123047167, −19.09479243746490395340537761169, −18.42639005729933440905667749370, −17.73349273420251700074867070805, −17.257281876215337821930388127313, −16.3177297612808944875141090537, −15.35333959136128721199366727698, −14.16399414518473815322957046043, −13.86612083690726214211516330228, −12.96892416815266605227789218796, −11.69018992646599310121845529380, −11.41683543702876020610880791226, −10.49813940720703554270855555910, −9.85980482076651293685024619587, −9.18385783629419677284274872310, −7.83241197966829960418498278812, −7.57324153239567372910425173340, −6.54056356132854755319453985541, −5.30450051703855390249663967299, −4.29097190471449190038105275228, −3.3549270456074898060470820361, −2.55365349922500269449028691170, −1.57503842286227981174486181674, −0.63357619797560798664625250319, 0.87109591548719805559254878494, 1.54189050491959559746010252970, 2.67905191824856983641005036688, 4.44361834207561184301632120231, 4.95070946065613582351208917139, 5.79754892774540787453735941740, 6.51545263618286248199727577294, 7.70048719574598986897886958020, 8.40177575830720714926046471107, 8.94722936360002385941868000064, 9.71472798090548413728528531225, 10.64284678169333728791373083197, 11.52907610090018763725674894025, 12.55528451706195197500855596928, 13.28901139325502918783994380588, 14.30330149522490019747254542129, 14.80778714531828114697181611600, 15.81493011855311634777485983358, 16.28190422543453193940411874825, 17.389385974543714959950229170483, 17.56693702785326268125412845528, 18.49055068718875577648163399726, 19.28543181147165167903072799733, 20.10829448650273174750710471630, 20.87695281002933374463899823424

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