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Properties

Label 1-45-45.4-r0-0-0
Degree $1$
Conductor $45$
Sign $0.173 - 0.984i$
Analytic cond. $0.208979$
Root an. cond. $0.208979$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s + 19-s + (0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + 26-s − 28-s + (−0.5 + 0.866i)29-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s + 19-s + (0.5 + 0.866i)22-s + (0.5 + 0.866i)23-s + 26-s − 28-s + (−0.5 + 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(0.208979\)
Root analytic conductor: \(0.208979\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 45,\ (0:\ ),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7698993003 - 0.6460222189i\)
\(L(\frac12)\) \(\approx\) \(0.7698993003 - 0.6460222189i\)
\(L(1)\) \(\approx\) \(1.017581897 - 0.5750616325i\)
\(L(1)\) \(\approx\) \(1.017581897 - 0.5750616325i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−34.6407752538296581875675095088, −33.45017423453234422579422601726, −32.34815851101980637749458547809, −31.28019394667002352521629623836, −30.40169614290229086472191382603, −28.76973431101492980994027655010, −27.323149853576368594941849747822, −26.292789499238961684723885885414, −24.89906916403401597878017476639, −24.26111227348730409067430609118, −22.81647949664846069167383640733, −21.78380647276188230474172592563, −20.64500112815866818819262540216, −18.595454476190985320021057180235, −17.663685799203339890063230028939, −16.086899998376124901253161876223, −15.223538835595963485719999274803, −13.85689829113214396106696827719, −12.65410094964942889104495091245, −11.157388556562852776233813918069, −8.94674731380379868426180375490, −7.90112655391428801320618356990, −6.129883513417807024140256986869, −4.986080434395568954318652182475, −3.036966339852487507043603700014, 1.79552054022879431760879518606, 3.82246261151636447130164381576, 5.12318840361089192244794614766, 7.13391008819698984650180343164, 9.164117984764452576349564931991, 10.5720086863329850717008945500, 11.62991026140771176012653565518, 13.16345457638311621835241422061, 14.12768025380590189428613636865, 15.54684695455579445900125710702, 17.42732327354157056402994216485, 18.61117112335464692242434707410, 20.09918331143247803638648325371, 20.7941643938762227796889730662, 22.16099686270101297940073783120, 23.370601086317253676682092149431, 24.20305329339874632344423794230, 26.10599953904193255973805087584, 27.304690595780937403189399033935, 28.51586223071429141720772834289, 29.476846827760444864649099007665, 30.77918517993744363590499976324, 31.33888421974699914084788635818, 33.125777829606636990567130411103, 33.468385069609883259802732782051

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