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Properties

Label 1-1375-1375.1009-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.0443 - 0.999i$
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  + (−0.929 + 0.368i)2-s + (−0.728 + 0.684i)3-s + (0.728 − 0.684i)4-s + (0.425 − 0.904i)6-s + 7-s + (−0.425 + 0.904i)8-s + (0.0627 − 0.998i)9-s + (−0.0627 + 0.998i)12-s + (0.0627 − 0.998i)13-s + (−0.929 + 0.368i)14-s + (0.0627 − 0.998i)16-s + (0.876 − 0.481i)17-s + (0.309 + 0.951i)18-s + (−0.876 + 0.481i)19-s + (−0.728 + 0.684i)21-s + ⋯
L(s)  = 1  + (−0.929 + 0.368i)2-s + (−0.728 + 0.684i)3-s + (0.728 − 0.684i)4-s + (0.425 − 0.904i)6-s + 7-s + (−0.425 + 0.904i)8-s + (0.0627 − 0.998i)9-s + (−0.0627 + 0.998i)12-s + (0.0627 − 0.998i)13-s + (−0.929 + 0.368i)14-s + (0.0627 − 0.998i)16-s + (0.876 − 0.481i)17-s + (0.309 + 0.951i)18-s + (−0.876 + 0.481i)19-s + (−0.728 + 0.684i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0443 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0443 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.0443 - 0.999i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.0443 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4437325794 - 0.4244552936i\)
\(L(\frac12)\) \(\approx\) \(0.4437325794 - 0.4244552936i\)
\(L(1)\) \(\approx\) \(0.5919265768 + 0.09652303473i\)
\(L(1)\) \(\approx\) \(0.5919265768 + 0.09652303473i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.929 + 0.368i)T \)
3 \( 1 + (-0.728 + 0.684i)T \)
7 \( 1 + T \)
13 \( 1 + (0.0627 - 0.998i)T \)
17 \( 1 + (0.876 - 0.481i)T \)
19 \( 1 + (-0.876 + 0.481i)T \)
23 \( 1 + (-0.535 + 0.844i)T \)
29 \( 1 + (0.187 - 0.982i)T \)
31 \( 1 + (-0.992 - 0.125i)T \)
37 \( 1 + (-0.535 - 0.844i)T \)
41 \( 1 + (0.929 + 0.368i)T \)
43 \( 1 + (0.309 - 0.951i)T \)
47 \( 1 + (0.992 - 0.125i)T \)
53 \( 1 + (0.425 + 0.904i)T \)
59 \( 1 + (-0.637 + 0.770i)T \)
61 \( 1 + (-0.968 - 0.248i)T \)
67 \( 1 + (0.425 - 0.904i)T \)
71 \( 1 + (-0.187 + 0.982i)T \)
73 \( 1 + (0.535 - 0.844i)T \)
79 \( 1 + (0.187 - 0.982i)T \)
83 \( 1 + (-0.187 - 0.982i)T \)
89 \( 1 + (0.0627 + 0.998i)T \)
97 \( 1 + (0.992 - 0.125i)T \)
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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

−20.83685159684492528692436536524, −19.85710331126568091132620455511, −19.14338899025525656720262770680, −18.457185403313841976048424316288, −17.97036802845689880077837306235, −17.07663681552127033988880980237, −16.71938818762948122422452881522, −15.86351967657094527910650057414, −14.69547123827903003509305967331, −13.95769849189370247022471517568, −12.72152235325536516908635502419, −12.29210988819644644904426992768, −11.41047499156669559567316057745, −10.89283133440522588788741596235, −10.20736394231169242390220821036, −8.989254914692247319846753356263, −8.31572708523872038524030746092, −7.546583647127223074727806061887, −6.79423490148068547790668381841, −6.00956875635399322370315526013, −4.88686375720251407263833071626, −3.93321912300783322227064681874, −2.47484050885798463695126805787, −1.73275787349873103467349168146, −1.00537484441386585117130261550, 0.217810998189998563933688292645, 1.12433595258929109662491121926, 2.233839795071147799952914096785, 3.566193286470968856695763745155, 4.64549270213813825502355833795, 5.71200838840276899325304974192, 5.85014225174688203231253991181, 7.3237958219248128253661134227, 7.842063882796303509406802129138, 8.82840577845223207518085321445, 9.59169249938972082789836141382, 10.52930852765800716582150258950, 10.815934511601414112187483672312, 11.810452945163319753326942983190, 12.37320397362015349707587113503, 13.88928842384978586581010111228, 14.72611174574805682383450121016, 15.324668921705758702029527759173, 15.97852771136149558771224730634, 16.9068683850586308786852032040, 17.33472544846751338956516492530, 18.06547961549150155038280371943, 18.62941277947167661311256005987, 19.727528343853169201272777834978, 20.55150516995923357866826978449

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