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Properties

Label 1-15e2-225.148-r1-0-0
Degree $1$
Conductor $225$
Sign $0.658 - 0.752i$
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  + (0.743 − 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

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

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.658 - 0.752i$
Analytic conductor: \(24.1796\)
Root analytic conductor: \(24.1796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (1:\ ),\ 0.658 - 0.752i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.135433087 - 1.422295527i\)
\(L(\frac12)\) \(\approx\) \(3.135433087 - 1.422295527i\)
\(L(1)\) \(\approx\) \(1.789707558 - 0.6328670412i\)
\(L(1)\) \(\approx\) \(1.789707558 - 0.6328670412i\)

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

−26.27650988506228817393350048441, −25.03495956260491122242654496007, −24.588389882607085182862066402579, −23.48593402281120990436521853557, −22.82677666668948754139476323277, −21.77216168685684206535211435440, −20.854172721030994665002590354497, −20.15124537699258647757945262065, −18.56734771277840706509569004606, −17.62113017054210509331831589631, −16.70770683073636409167089268363, −15.862865435436501426245470112242, −14.73079530622244096330376478123, −13.94176428829011620128826560980, −13.2230142532072497575701652240, −11.76131169979688391506071375514, −11.225069227779739062130254196489, −9.577574071206457166377126515273, −8.16650259057703265300862849536, −7.593710877528089564980977340615, −6.18946471263293782947744539917, −5.31279726136778999885039902049, −4.07187766951247884472218335978, −3.09339221491063056244683898275, −1.165538270737377302940271463722, 1.23397094026214031901278177599, 2.22915719526810062322487769686, 3.74108429594201585023854618801, 4.71279225708226543570710418979, 5.807839793127610815211638085159, 6.95689612206574930115283024088, 8.53744348559098609455845924460, 9.58392237787005834323053816262, 10.77907667722598842667310822792, 11.69746055536357865847357016511, 12.39568675866612771338068904978, 13.62329389032769030066182819965, 14.55537323570526365318792930684, 15.19632432552277229340344641471, 16.47985371612021677660576639060, 17.86532237354761326090695105523, 18.64413822101683090650947574448, 19.724486894384252388741720013807, 20.63715597463602196234951797121, 21.39425795135209283955782928374, 22.24847296057758746733018675322, 23.20026665671459928832047035302, 24.12977914240623043063960408079, 24.83242673490027010737334873863, 26.02037285197565458935697026718

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