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Properties

Label 1-117-117.25-r0-0-0
Degree $1$
Conductor $117$
Sign $0.173 + 0.984i$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s − 8-s + 10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + 17-s − 19-s + (0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s − 8-s + 10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + 17-s − 19-s + (0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(0.543345\)
Root analytic conductor: \(0.543345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (0:\ ),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.067948710 + 0.8961153693i\)
\(L(\frac12)\) \(\approx\) \(1.067948710 + 0.8961153693i\)
\(L(1)\) \(\approx\) \(1.191871764 + 0.6451893938i\)
\(L(1)\) \(\approx\) \(1.191871764 + 0.6451893938i\)

Euler product

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

−29.499470504477703539537007746276, −28.02174522029504583508004906949, −27.09623126728098279482566981458, −26.19582288668527249596570850746, −24.72769415731251479498117731387, −23.59525726883986215426958047008, −22.76713059156623032799907290111, −21.67356231273020342641088518300, −21.00848195346114083790464876638, −19.75899135564110679668101418911, −18.82874890362890044480236480716, −17.842221201341040499068273627601, −16.615607600147492164747546577363, −14.73127870187947901489553337384, −14.22761675717029177979384549568, −13.247001496982654437303459483844, −11.82746536198846904289823290767, −10.73854477163926637754467237274, −10.14712450231741440092506827758, −8.58873242245059907122150638241, −6.836717704166833946768117426090, −5.65677282027855632809497717644, −4.129741553478762654431907276196, −2.98920976045348041017487324351, −1.43953985584737901792371024827, 2.054907577994694077965041526842, 4.073241575106952478982921086752, 5.224491509196370250724627779077, 6.10145036462221690319129487602, 7.676982780911402790092696873171, 8.7417779599640911777791877740, 9.712525819362819759116194508044, 11.882769792167515423037970636908, 12.55042380166292883903458349953, 13.75684615488050219548132008498, 14.84138873071722722790727567427, 15.72925792225618222449042885613, 17.056601077828851735643016379203, 17.56400491254571950646571853644, 18.92136894714334107780198520218, 20.63329698210662452313619923808, 21.33025928897640237727114435916, 22.34690275381276980624988073729, 23.52788457900165204566823163208, 24.47455002281503740062649296105, 25.24701670664513226296121133885, 25.923486151924814166776918340942, 27.67397571080857909159941415437, 27.99402309699696102059370805482, 29.63134872813522645666369492977

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