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Properties

Label 1-15e2-225.191-r1-0-0
Degree $1$
Conductor $225$
Sign $0.752 - 0.658i$
Analytic cond. $24.1796$
Root an. cond. $24.1796$
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  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (−0.5 − 0.866i)7-s + (0.809 + 0.587i)8-s + (−0.669 + 0.743i)11-s + (0.669 + 0.743i)13-s + (0.978 + 0.207i)14-s + (−0.978 + 0.207i)16-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.104 − 0.994i)22-s + (0.978 + 0.207i)23-s − 26-s + (−0.809 + 0.587i)28-s + (−0.913 − 0.406i)29-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.104 − 0.994i)4-s + (−0.5 − 0.866i)7-s + (0.809 + 0.587i)8-s + (−0.669 + 0.743i)11-s + (0.669 + 0.743i)13-s + (0.978 + 0.207i)14-s + (−0.978 + 0.207i)16-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.104 − 0.994i)22-s + (0.978 + 0.207i)23-s − 26-s + (−0.809 + 0.587i)28-s + (−0.913 − 0.406i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.752 - 0.658i$
Analytic conductor: \(24.1796\)
Root analytic conductor: \(24.1796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (1:\ ),\ 0.752 - 0.658i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8288289777 - 0.3115363314i\)
\(L(\frac12)\) \(\approx\) \(0.8288289777 - 0.3115363314i\)
\(L(1)\) \(\approx\) \(0.7010338545 + 0.08979548832i\)
\(L(1)\) \(\approx\) \(0.7010338545 + 0.08979548832i\)

Euler product

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

−26.40221436341337738136780878945, −25.47563720110252524357846346618, −24.86907550509550082500348625725, −23.274018101636650875770979387642, −22.44998832839451683061523204284, −21.32593054470190859246674858349, −20.83567540084870990384007549981, −19.59842915431705328710803752249, −18.6408089245381618970683164475, −18.30000095353346319971676572099, −16.860813054850294135002298352919, −16.11733965546772811111073776012, −15.03206459080385310867727367099, −13.42242161331935718946534099998, −12.73991898606516912455725395654, −11.69341154621013474236157812213, −10.701339317213936013054425985463, −9.768525847542662985644152958402, −8.64786967165856323459112080530, −7.95270564617241186771724489721, −6.41886347555909417025604333526, −5.15503335439751852329996130710, −3.40755827688060971935992652732, −2.69539317018201980070458523679, −1.06046517386375320256953231498, 0.43794963953373668270761751287, 1.93205144966229896006586142880, 3.856843809622730789047490584309, 5.104314744343678034159911528845, 6.42640873669784300564838633694, 7.21211721676440928387120386508, 8.25439042764229289185260428306, 9.43097369680896437640932374047, 10.29696536380183279455890705174, 11.191271757700230289833692965084, 12.8817682439665305268910683255, 13.72887536923826414989186227004, 14.88528721097714429470758815941, 15.71929730598666318311958228343, 16.79695985806843355808882689874, 17.324351111847262038340591513141, 18.59980688065076901003420095095, 19.26638380440903986815927848768, 20.31300953666898644097754245433, 21.283059588660135098302560940233, 22.940887300671507842791619939707, 23.331441836687708661054097597556, 24.21995460987224446603176959188, 25.54780246643125555192671819430, 25.99447968214633203768114447613

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