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Properties

Label 1-135-135.94-r0-0-0
Degree $1$
Conductor $135$
Sign $0.549 - 0.835i$
Analytic cond. $0.626937$
Root an. cond. $0.626937$
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  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)11-s + (0.939 + 0.342i)13-s + (−0.939 − 0.342i)14-s + (0.173 − 0.984i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)22-s + (−0.766 + 0.642i)23-s + 26-s − 28-s + (−0.939 + 0.342i)29-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)7-s + (0.5 − 0.866i)8-s + (0.173 − 0.984i)11-s + (0.939 + 0.342i)13-s + (−0.939 − 0.342i)14-s + (0.173 − 0.984i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)22-s + (−0.766 + 0.642i)23-s + 26-s − 28-s + (−0.939 + 0.342i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.549 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(135\)    =    \(3^{3} \cdot 5\)
Sign: $0.549 - 0.835i$
Analytic conductor: \(0.626937\)
Root analytic conductor: \(0.626937\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{135} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 135,\ (0:\ ),\ 0.549 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.555564503 - 0.8387529217i\)
\(L(\frac12)\) \(\approx\) \(1.555564503 - 0.8387529217i\)
\(L(1)\) \(\approx\) \(1.563687257 - 0.5293386532i\)
\(L(1)\) \(\approx\) \(1.563687257 - 0.5293386532i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (-0.766 - 0.642i)T \)
11 \( 1 + (0.173 - 0.984i)T \)
13 \( 1 + (0.939 + 0.342i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 - T \)
59 \( 1 + (0.173 + 0.984i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (0.939 - 0.342i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.173 + 0.984i)T \)
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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

−28.73200223262989358639028264975, −27.979668232328718570897456765141, −26.31622005799306622878208679487, −25.48513035706342775462742944670, −24.86248947435391080825106243449, −23.518210381866351559916553560835, −22.753870354770786213227542538613, −21.96945667269252120414955809858, −20.79910394782676901171962608101, −19.95742195813367798338389886869, −18.578059012259897477596277779053, −17.36488342917830459119463159652, −16.07862692356473584877421020358, −15.46202990851435556245711464447, −14.3385857902149379434817322851, −13.14065938445111976794876623390, −12.382292244024776157757942800574, −11.28870222937521274113979922995, −9.80587244217098387929002896954, −8.41576677696786467599182719591, −7.02391011596366889460735246570, −6.079549191245975894344905203314, −4.85968535195818489237848362097, −3.511815514576040201246034653582, −2.26744732590397788980735884318, 1.467492042156706085352789117302, 3.33445443130058517532795827876, 4.02512549162021800493950069307, 5.81077433119848775058726942274, 6.507969081622171481385815747, 8.09417103166226051415016692144, 9.784087169209100010629372876291, 10.766791948319756148859230982055, 11.812936784567303671556120392186, 13.08865645058518956846691946774, 13.72791944231938541613570952264, 14.85858352820393595785354322554, 16.13266790160441027253796482430, 16.79685524419235199297392247683, 18.71600384708229460697212897688, 19.42243717235224041141323955758, 20.53286338813345872740134701182, 21.44654881009656617045344830952, 22.410403579622514660180181181, 23.41579275689285416333592865571, 24.04179139178740717991366725356, 25.37935874707098400596475608246, 26.20550552689218542367810393621, 27.6403342182807043731225192277, 28.6468977633722051699149577665

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