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Properties

Label 1-1375-1375.108-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.429 - 0.902i$
Analytic cond. $147.764$
Root an. cond. $147.764$
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  + (0.481 − 0.876i)2-s + (−0.844 + 0.535i)3-s + (−0.535 − 0.844i)4-s + (0.0627 + 0.998i)6-s − i·7-s + (−0.998 + 0.0627i)8-s + (0.425 − 0.904i)9-s + (0.904 + 0.425i)12-s + (−0.904 − 0.425i)13-s + (−0.876 − 0.481i)14-s + (−0.425 + 0.904i)16-s + (0.368 + 0.929i)17-s + (−0.587 − 0.809i)18-s + (0.929 − 0.368i)19-s + (0.535 + 0.844i)21-s + ⋯
L(s)  = 1  + (0.481 − 0.876i)2-s + (−0.844 + 0.535i)3-s + (−0.535 − 0.844i)4-s + (0.0627 + 0.998i)6-s − i·7-s + (−0.998 + 0.0627i)8-s + (0.425 − 0.904i)9-s + (0.904 + 0.425i)12-s + (−0.904 − 0.425i)13-s + (−0.876 − 0.481i)14-s + (−0.425 + 0.904i)16-s + (0.368 + 0.929i)17-s + (−0.587 − 0.809i)18-s + (0.929 − 0.368i)19-s + (0.535 + 0.844i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.429 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.429 - 0.902i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.429 - 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8897572123 - 1.409039758i\)
\(L(\frac12)\) \(\approx\) \(0.8897572123 - 1.409039758i\)
\(L(1)\) \(\approx\) \(0.8419545690 - 0.4954370818i\)
\(L(1)\) \(\approx\) \(0.8419545690 - 0.4954370818i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.481 - 0.876i)T \)
3 \( 1 + (-0.844 + 0.535i)T \)
7 \( 1 - iT \)
13 \( 1 + (-0.904 - 0.425i)T \)
17 \( 1 + (0.368 + 0.929i)T \)
19 \( 1 + (0.929 - 0.368i)T \)
23 \( 1 + (0.684 + 0.728i)T \)
29 \( 1 + (-0.968 + 0.248i)T \)
31 \( 1 + (-0.637 - 0.770i)T \)
37 \( 1 + (0.684 - 0.728i)T \)
41 \( 1 + (0.876 - 0.481i)T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (0.770 + 0.637i)T \)
53 \( 1 + (0.998 + 0.0627i)T \)
59 \( 1 + (0.992 + 0.125i)T \)
61 \( 1 + (-0.187 + 0.982i)T \)
67 \( 1 + (0.998 - 0.0627i)T \)
71 \( 1 + (0.968 - 0.248i)T \)
73 \( 1 + (0.684 + 0.728i)T \)
79 \( 1 + (-0.968 + 0.248i)T \)
83 \( 1 + (-0.248 + 0.968i)T \)
89 \( 1 + (0.425 + 0.904i)T \)
97 \( 1 + (0.770 + 0.637i)T \)
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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

−21.32456548485980052760346547982, −20.213879153138448341563171857625, −18.93335536077808803786403485230, −18.471073753081852253905210502152, −17.85297392851970231095932334350, −16.92769865307792435580121257353, −16.38828875471359361624732435540, −15.72806194610945628277198617122, −14.737875097738923899566872526531, −14.162053081635896119168399773229, −13.127863668420982382076181905194, −12.4981869710418909140329551049, −11.86574612212620957515057573596, −11.259646561491510985578084342846, −9.8113779540464742469995158090, −9.14104463894750576902888717475, −8.05076161633550838973114099907, −7.31617763972598283108429328190, −6.67382987539787700846522475906, −5.69134157116238785042038763528, −5.22654418553622520554591618420, −4.46014470382971651803316532011, −3.07397767861681929183060776238, −2.1887022070476060285589962562, −0.69570773726455273144844974326, 0.50077627060484919074307116815, 1.19179661930131205051637242066, 2.54294643672163981294021789414, 3.78406799403905631645079990886, 4.06147044204226732026318020538, 5.33506732253440818009121279854, 5.57775474816676354412587413563, 6.88604555037219830146251791228, 7.6706593159345434415152579640, 9.25011007945014094838726278987, 9.69391529273525522234086543844, 10.622871283951542037300269358810, 10.99464840088337922338052000077, 11.85720630279220700597125774469, 12.68619793118682838801631334767, 13.24949745214663705331699517969, 14.327511439880919307825371233135, 14.94300030707270695597277760937, 15.76642200260760864474487704606, 16.83074894994403116978108685308, 17.35608924101793318509105662387, 18.09592574704951578373103184927, 19.07178158120166794239422916102, 19.8779815479163960240021614529, 20.47068359956627227497128909135

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