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Properties

Label 1-675-675.302-r0-0-0
Degree $1$
Conductor $675$
Sign $0.861 + 0.508i$
Analytic cond. $3.13468$
Root an. cond. $3.13468$
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  + (−0.469 + 0.882i)2-s + (−0.559 − 0.829i)4-s + (−0.342 − 0.939i)7-s + (0.994 − 0.104i)8-s + (−0.848 + 0.529i)11-s + (0.469 + 0.882i)13-s + (0.990 + 0.139i)14-s + (−0.374 + 0.927i)16-s + (0.406 − 0.913i)17-s + (0.104 + 0.994i)19-s + (−0.0697 − 0.997i)22-s + (−0.788 − 0.615i)23-s − 26-s + (−0.587 + 0.809i)28-s + (−0.719 + 0.694i)29-s + ⋯
L(s)  = 1  + (−0.469 + 0.882i)2-s + (−0.559 − 0.829i)4-s + (−0.342 − 0.939i)7-s + (0.994 − 0.104i)8-s + (−0.848 + 0.529i)11-s + (0.469 + 0.882i)13-s + (0.990 + 0.139i)14-s + (−0.374 + 0.927i)16-s + (0.406 − 0.913i)17-s + (0.104 + 0.994i)19-s + (−0.0697 − 0.997i)22-s + (−0.788 − 0.615i)23-s − 26-s + (−0.587 + 0.809i)28-s + (−0.719 + 0.694i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.861 + 0.508i$
Analytic conductor: \(3.13468\)
Root analytic conductor: \(3.13468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (302, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ 0.861 + 0.508i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8804193334 + 0.2403050213i\)
\(L(\frac12)\) \(\approx\) \(0.8804193334 + 0.2403050213i\)
\(L(1)\) \(\approx\) \(0.7508168775 + 0.2118892888i\)
\(L(1)\) \(\approx\) \(0.7508168775 + 0.2118892888i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.469 + 0.882i)T \)
7 \( 1 + (-0.342 - 0.939i)T \)
11 \( 1 + (-0.848 + 0.529i)T \)
13 \( 1 + (0.469 + 0.882i)T \)
17 \( 1 + (0.406 - 0.913i)T \)
19 \( 1 + (0.104 + 0.994i)T \)
23 \( 1 + (-0.788 - 0.615i)T \)
29 \( 1 + (-0.719 + 0.694i)T \)
31 \( 1 + (0.961 - 0.275i)T \)
37 \( 1 + (0.743 + 0.669i)T \)
41 \( 1 + (0.882 - 0.469i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (0.275 - 0.961i)T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (0.848 + 0.529i)T \)
61 \( 1 + (0.0348 + 0.999i)T \)
67 \( 1 + (0.694 - 0.719i)T \)
71 \( 1 + (0.104 - 0.994i)T \)
73 \( 1 + (0.743 - 0.669i)T \)
79 \( 1 + (0.719 - 0.694i)T \)
83 \( 1 + (-0.898 + 0.438i)T \)
89 \( 1 + (0.669 + 0.743i)T \)
97 \( 1 + (0.970 - 0.241i)T \)
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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

−22.48473299805601344401354533869, −21.56254283163451259329121641745, −21.22537165096542969410771533277, −20.133443584826854375746038152002, −19.37557447637925845525273219984, −18.68338050197169335835340481983, −17.93880534724909680691369386478, −17.25165519074955382013755234445, −15.987869772136604971173860227412, −15.53991539333910140745485538893, −14.22130224931993291230199043471, −13.02529674563971015730701452059, −12.822560883603400056979109379525, −11.62995852091689721468178776931, −10.95180533928793509708051083701, −10.02932479217277993803989879295, −9.236236344713649251022877817168, −8.26173072810605222073687716897, −7.73419557288278817844219452005, −6.11967744668878109288352562755, −5.34034365264499161567200736848, −4.01860571607189903508479927388, −2.98537421249473807726627162425, −2.319992979152201061152045202536, −0.85895130314335339193093774411, 0.76038453361757374079247475227, 2.10705968122783193277985003339, 3.754294560978448491549654335572, 4.617646086206680755185435538954, 5.6703735295256804457724549321, 6.650878807908376686017409598274, 7.41760954366524347799709603112, 8.12867361521034270592103385573, 9.27020564396483281598015153234, 10.06146696642262629049168593704, 10.672058192516090850257781308850, 11.938968261619597635919680649713, 13.136638342964944863972491954455, 13.863424508139250828168980059924, 14.52708785644968166956482412810, 15.611292803947042901992322237492, 16.416595108146872082523966478933, 16.770432928516290198240415228311, 17.99271527184586745924346878725, 18.49675166674659074392936546443, 19.35762887437169594209783145978, 20.378226095940763670363159426628, 20.95199252408038923520517975679, 22.44800983325245484051848621657, 22.95780279304789501452645659621

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