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Properties

Label 1-15e2-225.23-r0-0-0
Degree $1$
Conductor $225$
Sign $0.752 + 0.658i$
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  + (−0.743 + 0.669i)2-s + (0.104 − 0.994i)4-s + (0.866 + 0.5i)7-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)11-s + (0.743 + 0.669i)13-s + (−0.978 + 0.207i)14-s + (−0.978 − 0.207i)16-s + (−0.587 − 0.809i)17-s + (0.809 − 0.587i)19-s + (0.994 + 0.104i)22-s + (0.207 + 0.978i)23-s − 26-s + (0.587 − 0.809i)28-s + (0.913 − 0.406i)29-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)2-s + (0.104 − 0.994i)4-s + (0.866 + 0.5i)7-s + (0.587 + 0.809i)8-s + (−0.669 − 0.743i)11-s + (0.743 + 0.669i)13-s + (−0.978 + 0.207i)14-s + (−0.978 − 0.207i)16-s + (−0.587 − 0.809i)17-s + (0.809 − 0.587i)19-s + (0.994 + 0.104i)22-s + (0.207 + 0.978i)23-s − 26-s + (0.587 − 0.809i)28-s + (0.913 − 0.406i)29-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.752 + 0.658i$
Analytic conductor: \(1.04489\)
Root analytic conductor: \(1.04489\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (0:\ ),\ 0.752 + 0.658i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8488185722 + 0.3190499260i\)
\(L(\frac12)\) \(\approx\) \(0.8488185722 + 0.3190499260i\)
\(L(1)\) \(\approx\) \(0.8069968675 + 0.2230928026i\)
\(L(1)\) \(\approx\) \(0.8069968675 + 0.2230928026i\)

Euler product

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

−26.573357921319956888104169758400, −25.61112202021257739031955204967, −24.67661813357238384456935341989, −23.46739353447272143020886622286, −22.508731727117822742412326710598, −21.33190083725166729379383737242, −20.52010484070339297869414340126, −20.026622639302535484677092385794, −18.63036660283134107440648947468, −17.94489840870123483508937957754, −17.20306438838390325280312257386, −16.06650551529967666728130693520, −14.99405836973275149028665645408, −13.61417549248842427420108825016, −12.71688005382186328814759277765, −11.62689463729192156401738683386, −10.61299537798038907819971974342, −10.02370941946089512510273479599, −8.46763091242634646143115507595, −7.94138276088451450628678803952, −6.69415215200493485188972893412, −4.94631876130650044531690961900, −3.79733573301774598821493468957, −2.39568737288646232819733855082, −1.137449416942544325269671568944, 1.198118421616263862566927518756, 2.67265698853906132271029857330, 4.689731308772326876394281408664, 5.608576458488204299131445697919, 6.76549618523614116085378915575, 7.949846682062670751026547555119, 8.71507738296361294719651365974, 9.695970106940171252597418750739, 11.09601406526017012321322889569, 11.57943539177267406180939125434, 13.52497078600676611916530561619, 14.14513688298670492412968059384, 15.54196034994034175073999214694, 15.90718312752746719024994049732, 17.19763987927005146517534965354, 18.125382125034100114188491642015, 18.66656090130776283851424588541, 19.808828094044789660174396473591, 20.88812568613565826973626152519, 21.79396123589173299128368567532, 23.22909464538177409993989647430, 23.946775703294683196541787200542, 24.75388453571320313882840460910, 25.57985645764414420304473704635, 26.77565013011698301534395173008

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