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Properties

Label 1-15e2-225.212-r0-0-0
Degree $1$
Conductor $225$
Sign $-0.152 + 0.988i$
Analytic cond. $1.04489$
Root an. cond. $1.04489$
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  + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.866 − 0.5i)7-s + (0.587 − 0.809i)8-s + (0.978 + 0.207i)11-s + (0.207 + 0.978i)13-s + (0.669 − 0.743i)14-s + (0.669 + 0.743i)16-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.406 + 0.913i)22-s + (0.743 + 0.669i)23-s − 26-s + (0.587 + 0.809i)28-s + (−0.104 − 0.994i)29-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (−0.913 − 0.406i)4-s + (−0.866 − 0.5i)7-s + (0.587 − 0.809i)8-s + (0.978 + 0.207i)11-s + (0.207 + 0.978i)13-s + (0.669 − 0.743i)14-s + (0.669 + 0.743i)16-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.406 + 0.913i)22-s + (0.743 + 0.669i)23-s − 26-s + (0.587 + 0.809i)28-s + (−0.104 − 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.152 + 0.988i$
Analytic conductor: \(1.04489\)
Root analytic conductor: \(1.04489\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (0:\ ),\ -0.152 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5805480182 + 0.6773369950i\)
\(L(\frac12)\) \(\approx\) \(0.5805480182 + 0.6773369950i\)
\(L(1)\) \(\approx\) \(0.7427610332 + 0.4306596532i\)
\(L(1)\) \(\approx\) \(0.7427610332 + 0.4306596532i\)

Euler product

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

−26.37268128158501383909702270398, −25.36767574241085795620473895777, −24.449302414821620089948630324911, −22.816570997371032000954474304963, −22.46365903298824020047801910833, −21.583624613050628394105074719790, −20.28912079188145034271360272672, −19.80356958439861071684190763932, −18.723194366332730756251976087546, −17.9797152275590549957476057237, −16.87401162381319117032926088228, −15.82568555800332531671473741458, −14.54195067690820194463186862631, −13.37232399471304333460135099980, −12.638576710676543398345357429077, −11.61279397820728315494222266719, −10.71741002101525767222128345127, −9.40084615961974156085101597628, −8.99319222633624705714121640178, −7.5110055782194736956960511704, −6.097660134001581595237723517753, −4.78105940601273917476192591821, −3.41611134631763690699483329948, −2.581741465053474539489221745684, −0.833233521894645178669906503592, 1.35401607616184432282186002599, 3.587802546418556190514913925829, 4.50640219771536582231251343393, 6.0770463075622464569289550383, 6.72382258141224733567795197000, 7.78257803572970128611806102385, 9.15614394032867867283896468096, 9.676743397581719690754689134394, 11.04403039424980356816273769239, 12.47888470208209734918811306717, 13.544770574334838361587318841829, 14.31477147775614026888740357860, 15.39915657212014120879678739489, 16.38999882681447344324376082405, 17.01276621333804485734523577681, 18.01068189861081742957969539798, 19.25867816954373148325301241853, 19.68839437753330740458917496230, 21.28956975917728477654301109913, 22.4059010742107146524071142878, 23.029542620422784703468415677415, 24.007634594715084727092036308853, 24.90672504224214115591709482729, 25.7839763610673974556172307447, 26.54391268067715958374417201235

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