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Properties

Label 1-117-117.88-r0-0-0
Degree $1$
Conductor $117$
Sign $0.329 - 0.944i$
Analytic cond. $0.543345$
Root an. cond. $0.543345$
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  − 2-s + 4-s + (0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)10-s − 11-s + (−0.5 + 0.866i)14-s + 16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + 22-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  − 2-s + 4-s + (0.5 − 0.866i)5-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)10-s − 11-s + (−0.5 + 0.866i)14-s + 16-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)20-s + 22-s + (−0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.329 - 0.944i$
Analytic conductor: \(0.543345\)
Root analytic conductor: \(0.543345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (0:\ ),\ 0.329 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5796762279 - 0.4114578188i\)
\(L(\frac12)\) \(\approx\) \(0.5796762279 - 0.4114578188i\)
\(L(1)\) \(\approx\) \(0.7092699010 - 0.2267519843i\)
\(L(1)\) \(\approx\) \(0.7092699010 - 0.2267519843i\)

Euler product

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

−29.06874824030783018515202442893, −28.47364045285981586296768368978, −27.31818099385370740741912031246, −26.30293070606599514552245353847, −25.65119502311471869522882759798, −24.60275789225958345901775479904, −23.54447019689352201717026088786, −21.82127545740787515605677614198, −21.33975384624290750247933491110, −19.94130576614444899311202341081, −18.81361674172319865328125592012, −18.02924852712577075694949914489, −17.38709522463510582767172881979, −15.687611259693426296249187743336, −15.16060280742114875943835814568, −13.701823107381087366587502611256, −12.10787711227254397777937218147, −10.99220113796771747247749752057, −10.13828059932805095460423935877, −8.89557894777247214551830023223, −7.797353676608832800761329441948, −6.54725923660207196821206199682, −5.3929307898028770419813858116, −2.957671162944370422656131989868, −1.9464847311208850498988523594, 0.96722574006208369688614514348, 2.47751335124447526450463857573, 4.57158617402300030878573775488, 5.9976684730721758651348408389, 7.529307090554019622666298184262, 8.3705480023184932426728462853, 9.67640526789122195707295829392, 10.527056640093139882455778361234, 11.79035973439861667089923972506, 13.083074599501985622534317830641, 14.301794054680448698094283486921, 15.91130960849135219835395009035, 16.587174992380849638670177424153, 17.68385206420357896302917487082, 18.38422685787995329842808849420, 19.93623561361531419322343018992, 20.59033421445679688789264542077, 21.33553401341923750881139611087, 23.15828858195482299104115506599, 24.30214033690693125847026063561, 24.92520703740009395844833452952, 26.20916663860709321089521826557, 26.92457716132580919572768653044, 28.016937013177798031594682429436, 28.93776442838861209209055642977

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