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Properties

Label 1-1287-1287.1237-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.977 + 0.209i$
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  + (0.406 − 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.994 + 0.104i)5-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)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 + (−0.743 − 0.669i)19-s + (−0.587 − 0.809i)20-s − 23-s + (0.978 + 0.207i)25-s + (−0.406 − 0.913i)28-s + (−0.978 + 0.207i)29-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)2-s + (−0.669 − 0.743i)4-s + (0.994 + 0.104i)5-s + (0.951 + 0.309i)7-s + (−0.951 + 0.309i)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 + (−0.743 − 0.669i)19-s + (−0.587 − 0.809i)20-s − 23-s + (0.978 + 0.207i)25-s + (−0.406 − 0.913i)28-s + (−0.978 + 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.977 + 0.209i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.977 + 0.209i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2013487997 - 1.899836529i\)
\(L(\frac12)\) \(\approx\) \(-0.2013487997 - 1.899836529i\)
\(L(1)\) \(\approx\) \(1.096769000 - 0.7964946592i\)
\(L(1)\) \(\approx\) \(1.096769000 - 0.7964946592i\)

Euler product

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

−21.15317387417900339358833526446, −20.985099683018901805130016445782, −19.75949235574458071121556034702, −18.58639773144565512807564456713, −17.934469165159173513491409921314, −17.24083505799589739345382617200, −16.83463560258784769713023878755, −15.86420581380807615179377071589, −14.88600338824255389840737257491, −14.31498608091502737082325926466, −13.800891658282366252011922860783, −12.83443077394933393036750638730, −12.28725024278342665782181262245, −11.0487814381118891434355410908, −10.21806096735670549831053196311, −9.309623192823783829921471598289, −8.36440671180933715163768037063, −7.871559648704332634285747474425, −6.746577423271153588213895382008, −6.00281604174079289495570253953, −5.32591362362633380795621833692, −4.42964142425114529200020072734, −3.6335705612203531234874635069, −2.24411297415493932629215676285, −1.34231033606521590646108758447, 0.28965608633931444143879983680, 1.56673062061857416997470129953, 2.17010445306420621413027668181, 3.008375043510537281985613468217, 4.265611106369172702415526972682, 5.03125046228619825278705801952, 5.71244793491316611888909712795, 6.60392829326976949669501340144, 7.88800438803851610230553716703, 8.9178523845402462240121463489, 9.496056589308312084982780764093, 10.38451757431230707745682755020, 11.09544165718249815972981160012, 11.81571049202313231693390984746, 12.634172456652343681803275343714, 13.593772040245848007429858962640, 13.97403808507193919020835931826, 14.82904784323030294213022094490, 15.49744679011196053500566950930, 16.89763548631937686151927339952, 17.53914123119106455785704653140, 18.399501474086483628011014639464, 18.653809501716167245445372519557, 19.93336680899286812904441594027, 20.53584777471915417989332931435

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