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Properties

Label 1-15e2-225.131-r1-0-0
Degree $1$
Conductor $225$
Sign $0.0279 - 0.999i$
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  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.5 + 0.866i)7-s + (−0.309 − 0.951i)8-s + (0.104 + 0.994i)11-s + (−0.104 + 0.994i)13-s + (−0.913 − 0.406i)14-s + (0.913 − 0.406i)16-s + (−0.309 − 0.951i)17-s + (0.309 + 0.951i)19-s + (−0.978 + 0.207i)22-s + (−0.913 − 0.406i)23-s − 26-s + (0.309 − 0.951i)28-s + (−0.669 − 0.743i)29-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.5 + 0.866i)7-s + (−0.309 − 0.951i)8-s + (0.104 + 0.994i)11-s + (−0.104 + 0.994i)13-s + (−0.913 − 0.406i)14-s + (0.913 − 0.406i)16-s + (−0.309 − 0.951i)17-s + (0.309 + 0.951i)19-s + (−0.978 + 0.207i)22-s + (−0.913 − 0.406i)23-s − 26-s + (0.309 − 0.951i)28-s + (−0.669 − 0.743i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0279 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0279 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.0279 - 0.999i$
Analytic conductor: \(24.1796\)
Root analytic conductor: \(24.1796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (1:\ ),\ 0.0279 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1586150702 + 0.1542464159i\)
\(L(\frac12)\) \(\approx\) \(-0.1586150702 + 0.1542464159i\)
\(L(1)\) \(\approx\) \(0.5872521339 + 0.4849070819i\)
\(L(1)\) \(\approx\) \(0.5872521339 + 0.4849070819i\)

Euler product

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

−25.83595612568122908726112039847, −24.24368174489577774270069407200, −23.559361206260655788533634556631, −22.42211916958948125101640126700, −21.86830705393099189915904442892, −20.717107028168107509317785389410, −19.7794057681740390094505810529, −19.32834408397183186005700980210, −18.006212107200624718615606666224, −17.24174381430913454769976626484, −16.04940420227411452817374812860, −14.747227120236828815731138941439, −13.57227130414222978128331560796, −13.10483424691125087061444878514, −11.87649382085844908182348369101, −10.780003272279051110480352668600, −10.17021003945625421685297638650, −8.94339234215053091244883376296, −7.89780585719432164649916324847, −6.34617810522194979013163409468, −5.10533104247194996113363488560, −3.77157193720571431067832200794, −2.99477374238872101070896202112, −1.29929584942984277398365135313, −0.07410075663743001711760224565, 2.14891566385379764072580379843, 3.81891508474146923385117539766, 4.931210023103798380026631048061, 6.07261604727228579398433949628, 6.96771985104138048977089836126, 8.08009444247137747130805604302, 9.30052367674607863623016609137, 9.84113896806977674867999848982, 11.79441767334967875095742814012, 12.52961528090618600290061168867, 13.73244775987535825829831689267, 14.616946513608192742108254854033, 15.60082111816342452181959206287, 16.299303221945247506262624764410, 17.35574726168016477537851185868, 18.392819563585032364771817368935, 19.01078292412339853770482235648, 20.44507739902510988246656000043, 21.59955537408274080716024597826, 22.55256323307863544631712919817, 23.071734864190983343736167366517, 24.44555964447073891833881521184, 24.897668836065125238592198218516, 25.96129899712969732489054853499, 26.55450221288298496996628148421

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